Metamath Proof Explorer

Theorem itg2addnclem

Description: An alternate expression for the S.2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017) (Revised by Brendan Leahy, 10-Mar-2018)

Ref Expression
Hypothesis itg2addnclem.1 
|- L = { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) }
Assertion itg2addnclem 
|- ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) = sup ( L , RR* , < ) )

Proof

Step Hyp Ref Expression
1 itg2addnclem.1 
 |-  L = { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) }
2 eqid 
 |-  { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } = { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) }
3 2 itg2val 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) = sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) )
4 1 supeq1i 
 |-  sup ( L , RR* , < ) = sup ( { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } , RR* , < )
5 xrltso 
 |-  < Or RR*
6 5 a1i 
 |-  ( F : RR --> ( 0 [,] +oo ) -> < Or RR* )
7 simprr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> x = ( S.1 ` f ) )
8 itg1cl 
 |-  ( f e. dom S.1 -> ( S.1 ` f ) e. RR )
9 8 rexrd 
 |-  ( f e. dom S.1 -> ( S.1 ` f ) e. RR* )
10 9 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> ( S.1 ` f ) e. RR* )
11 7 10 eqeltrd 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> x e. RR* )
12 11 rexlimiva 
 |-  ( E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) -> x e. RR* )
13 12 abssi 
 |-  { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } C_ RR*
14 supxrcl 
 |-  ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } C_ RR* -> sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) e. RR* )
15 13 14 mp1i 
 |-  ( F : RR --> ( 0 [,] +oo ) -> sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) e. RR* )
16 fveq1 
 |-  ( g = f -> ( g ` z ) = ( f ` z ) )
17 16 eqeq1d 
 |-  ( g = f -> ( ( g ` z ) = 0 <-> ( f ` z ) = 0 ) )
18 16 oveq1d 
 |-  ( g = f -> ( ( g ` z ) + y ) = ( ( f ` z ) + y ) )
19 17 18 ifbieq2d 
 |-  ( g = f -> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) = if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) )
20 19 mpteq2dv 
 |-  ( g = f -> ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) = ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) )
21 20 breq1d 
 |-  ( g = f -> ( ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F ) )
22 21 rexbidv 
 |-  ( g = f -> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F ) )
23 fveq2 
 |-  ( g = f -> ( S.1 ` g ) = ( S.1 ` f ) )
24 23 eqeq2d 
 |-  ( g = f -> ( x = ( S.1 ` g ) <-> x = ( S.1 ` f ) ) )
25 22 24 anbi12d 
 |-  ( g = f -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) <-> ( E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` f ) ) ) )
26 25 cbvrexvw 
 |-  ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) <-> E. f e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` f ) ) )
27 breq2 
 |-  ( 0 = if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) -> ( ( f ` z ) <_ 0 <-> ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) )
28 breq2 
 |-  ( ( ( f ` z ) + y ) = if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) -> ( ( f ` z ) <_ ( ( f ` z ) + y ) <-> ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) )
29 id 
 |-  ( ( f ` z ) = 0 -> ( f ` z ) = 0 )
30 0le0 
 |-  0 <_ 0
31 29 30 eqbrtrdi 
 |-  ( ( f ` z ) = 0 -> ( f ` z ) <_ 0 )
32 31 adantl 
 |-  ( ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) /\ ( f ` z ) = 0 ) -> ( f ` z ) <_ 0 )
33 rpge0 
 |-  ( y e. RR+ -> 0 <_ y )
34 33 ad2antlr 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> 0 <_ y )
35 i1ff 
 |-  ( f e. dom S.1 -> f : RR --> RR )
36 35 ffvelcdmda 
 |-  ( ( f e. dom S.1 /\ z e. RR ) -> ( f ` z ) e. RR )
37 36 adantlr 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) e. RR )
38 rpre 
 |-  ( y e. RR+ -> y e. RR )
39 38 ad2antlr 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> y e. RR )
40 37 39 addge01d 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> ( 0 <_ y <-> ( f ` z ) <_ ( ( f ` z ) + y ) ) )
41 34 40 mpbid 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) <_ ( ( f ` z ) + y ) )
42 41 adantr 
 |-  ( ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) /\ -. ( f ` z ) = 0 ) -> ( f ` z ) <_ ( ( f ` z ) + y ) )
43 27 28 32 42 ifbothda 
 |-  ( ( ( f e. dom S.1 /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) )
44 43 adantlll 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) )
45 35 ad2antlr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> f : RR --> RR )
46 45 ffvelcdmda 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) e. RR )
47 46 rexrd 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( f ` z ) e. RR* )
48 0re 
 |-  0 e. RR
49 38 ad2antlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> y e. RR )
50 46 49 readdcld 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( ( f ` z ) + y ) e. RR )
51 ifcl 
 |-  ( ( 0 e. RR /\ ( ( f ` z ) + y ) e. RR ) -> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) e. RR )
52 48 50 51 sylancr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) e. RR )
53 52 rexrd 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) e. RR* )
54 iccssxr 
 |-  ( 0 [,] +oo ) C_ RR*
55 fss 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( 0 [,] +oo ) C_ RR* ) -> F : RR --> RR* )
56 54 55 mpan2 
 |-  ( F : RR --> ( 0 [,] +oo ) -> F : RR --> RR* )
57 56 ad2antrr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> F : RR --> RR* )
58 57 ffvelcdmda 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( F ` z ) e. RR* )
59 xrletr 
 |-  ( ( ( f ` z ) e. RR* /\ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) e. RR* /\ ( F ` z ) e. RR* ) -> ( ( ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) /\ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) <_ ( F ` z ) ) -> ( f ` z ) <_ ( F ` z ) ) )
60 47 53 58 59 syl3anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( ( ( f ` z ) <_ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) /\ if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) <_ ( F ` z ) ) -> ( f ` z ) <_ ( F ` z ) ) )
61 44 60 mpand 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) /\ z e. RR ) -> ( if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) <_ ( F ` z ) -> ( f ` z ) <_ ( F ` z ) ) )
62 61 ralimdva 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> ( A. z e. RR if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) <_ ( F ` z ) -> A. z e. RR ( f ` z ) <_ ( F ` z ) ) )
63 reex 
 |-  RR e. _V
64 63 a1i 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> RR e. _V )
65 eqidd 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) = ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) )
66 id 
 |-  ( F : RR --> ( 0 [,] +oo ) -> F : RR --> ( 0 [,] +oo ) )
67 66 feqmptd 
 |-  ( F : RR --> ( 0 [,] +oo ) -> F = ( z e. RR |-> ( F ` z ) ) )
68 67 ad2antrr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> F = ( z e. RR |-> ( F ` z ) ) )
69 64 52 58 65 68 ofrfval2 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> ( ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F <-> A. z e. RR if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) <_ ( F ` z ) ) )
70 35 feqmptd 
 |-  ( f e. dom S.1 -> f = ( z e. RR |-> ( f ` z ) ) )
71 70 ad2antlr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> f = ( z e. RR |-> ( f ` z ) ) )
72 64 46 58 71 68 ofrfval2 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> ( f oR <_ F <-> A. z e. RR ( f ` z ) <_ ( F ` z ) ) )
73 62 69 72 3imtr4d 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ y e. RR+ ) -> ( ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F -> f oR <_ F ) )
74 73 rexlimdva 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) -> ( E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F -> f oR <_ F ) )
75 74 anim1d 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` f ) ) -> ( f oR <_ F /\ x = ( S.1 ` f ) ) ) )
76 75 reximdva 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( E. f e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( f ` z ) = 0 , 0 , ( ( f ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` f ) ) -> E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) ) )
77 26 76 biimtrid 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) -> E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) ) )
78 77 ss2abdv 
 |-  ( F : RR --> ( 0 [,] +oo ) -> { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } C_ { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } )
79 78 sseld 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( b e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } -> b e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } ) )
80 simp3r 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> x = ( S.1 ` f ) )
81 9 3ad2ant2 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> ( S.1 ` f ) e. RR* )
82 80 81 eqeltrd 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 /\ ( f oR <_ F /\ x = ( S.1 ` f ) ) ) -> x e. RR* )
83 82 rexlimdv3a 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) -> x e. RR* ) )
84 83 abssdv 
 |-  ( F : RR --> ( 0 [,] +oo ) -> { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } C_ RR* )
85 xrsupss 
 |-  ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } C_ RR* -> E. a e. RR* ( A. b e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } -. a < b /\ A. b e. RR* ( b < a -> E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s ) ) )
86 84 85 syl 
 |-  ( F : RR --> ( 0 [,] +oo ) -> E. a e. RR* ( A. b e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } -. a < b /\ A. b e. RR* ( b < a -> E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s ) ) )
87 6 86 supub 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( b e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } -> -. sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) < b ) )
88 79 87 syld 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( b e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } -> -. sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) < b ) )
89 88 imp 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } ) -> -. sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) < b )
90 supxrlub 
 |-  ( ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } C_ RR* /\ b e. RR* ) -> ( b < sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) <-> E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s ) )
91 13 90 mpan 
 |-  ( b e. RR* -> ( b < sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) <-> E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s ) )
92 91 adantl 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( b < sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) <-> E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s ) )
93 simprrr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> s = ( S.1 ` f ) )
94 93 breq2d 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( b < s <-> b < ( S.1 ` f ) ) )
95 simplll 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> F : RR --> ( 0 [,] +oo ) )
96 i1f0 
 |-  ( RR X. { 0 } ) e. dom S.1
97 2rp 
 |-  2 e. RR+
98 97 ne0ii 
 |-  RR+ =/= (/)
99 ffvelcdm 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ z e. RR ) -> ( F ` z ) e. ( 0 [,] +oo ) )
100 elxrge0 
 |-  ( ( F ` z ) e. ( 0 [,] +oo ) <-> ( ( F ` z ) e. RR* /\ 0 <_ ( F ` z ) ) )
101 99 100 sylib 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ z e. RR ) -> ( ( F ` z ) e. RR* /\ 0 <_ ( F ` z ) ) )
102 101 simprd 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ z e. RR ) -> 0 <_ ( F ` z ) )
103 102 ralrimiva 
 |-  ( F : RR --> ( 0 [,] +oo ) -> A. z e. RR 0 <_ ( F ` z ) )
104 63 a1i 
 |-  ( F : RR --> ( 0 [,] +oo ) -> RR e. _V )
105 c0ex 
 |-  0 e. _V
106 105 a1i 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ z e. RR ) -> 0 e. _V )
107 eqidd 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( z e. RR |-> 0 ) = ( z e. RR |-> 0 ) )
108 104 106 99 107 67 ofrfval2 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( ( z e. RR |-> 0 ) oR <_ F <-> A. z e. RR 0 <_ ( F ` z ) ) )
109 103 108 mpbird 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( z e. RR |-> 0 ) oR <_ F )
110 109 ralrimivw 
 |-  ( F : RR --> ( 0 [,] +oo ) -> A. y e. RR+ ( z e. RR |-> 0 ) oR <_ F )
111 r19.2z 
 |-  ( ( RR+ =/= (/) /\ A. y e. RR+ ( z e. RR |-> 0 ) oR <_ F ) -> E. y e. RR+ ( z e. RR |-> 0 ) oR <_ F )
112 98 110 111 sylancr 
 |-  ( F : RR --> ( 0 [,] +oo ) -> E. y e. RR+ ( z e. RR |-> 0 ) oR <_ F )
113 fveq2 
 |-  ( g = ( RR X. { 0 } ) -> ( S.1 ` g ) = ( S.1 ` ( RR X. { 0 } ) ) )
114 itg10 
 |-  ( S.1 ` ( RR X. { 0 } ) ) = 0
115 113 114 eqtr2di 
 |-  ( g = ( RR X. { 0 } ) -> 0 = ( S.1 ` g ) )
116 115 biantrud 
 |-  ( g = ( RR X. { 0 } ) -> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) ) )
117 fveq1 
 |-  ( g = ( RR X. { 0 } ) -> ( g ` z ) = ( ( RR X. { 0 } ) ` z ) )
118 105 fvconst2 
 |-  ( z e. RR -> ( ( RR X. { 0 } ) ` z ) = 0 )
119 117 118 sylan9eq 
 |-  ( ( g = ( RR X. { 0 } ) /\ z e. RR ) -> ( g ` z ) = 0 )
120 iftrue 
 |-  ( ( g ` z ) = 0 -> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) = 0 )
121 119 120 syl 
 |-  ( ( g = ( RR X. { 0 } ) /\ z e. RR ) -> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) = 0 )
122 121 mpteq2dva 
 |-  ( g = ( RR X. { 0 } ) -> ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) = ( z e. RR |-> 0 ) )
123 122 breq1d 
 |-  ( g = ( RR X. { 0 } ) -> ( ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> ( z e. RR |-> 0 ) oR <_ F ) )
124 123 rexbidv 
 |-  ( g = ( RR X. { 0 } ) -> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> E. y e. RR+ ( z e. RR |-> 0 ) oR <_ F ) )
125 116 124 bitr3d 
 |-  ( g = ( RR X. { 0 } ) -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) <-> E. y e. RR+ ( z e. RR |-> 0 ) oR <_ F ) )
126 125 rspcev 
 |-  ( ( ( RR X. { 0 } ) e. dom S.1 /\ E. y e. RR+ ( z e. RR |-> 0 ) oR <_ F ) -> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) )
127 96 112 126 sylancr 
 |-  ( F : RR --> ( 0 [,] +oo ) -> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) )
128 id 
 |-  ( b = -oo -> b = -oo )
129 mnflt 
 |-  ( 0 e. RR -> -oo < 0 )
130 48 129 mp1i 
 |-  ( b = -oo -> -oo < 0 )
131 128 130 eqbrtrd 
 |-  ( b = -oo -> b < 0 )
132 eqeq1 
 |-  ( a = 0 -> ( a = ( S.1 ` g ) <-> 0 = ( S.1 ` g ) ) )
133 132 anbi2d 
 |-  ( a = 0 -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) <-> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) ) )
134 133 rexbidv 
 |-  ( a = 0 -> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) <-> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) ) )
135 breq2 
 |-  ( a = 0 -> ( b < a <-> b < 0 ) )
136 134 135 anbi12d 
 |-  ( a = 0 -> ( ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) <-> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) /\ b < 0 ) ) )
137 105 136 spcev 
 |-  ( ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) /\ b < 0 ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
138 127 131 137 syl2an 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b = -oo ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
139 95 138 sylan 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b = -oo ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
140 simp-4r 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> b e. RR* )
141 8 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> ( S.1 ` f ) e. RR )
142 141 ad3antlr 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> ( S.1 ` f ) e. RR )
143 simpllr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> b e. RR* )
144 ngtmnft 
 |-  ( b e. RR* -> ( b = -oo <-> -. -oo < b ) )
145 144 biimprd 
 |-  ( b e. RR* -> ( -. -oo < b -> b = -oo ) )
146 145 necon1ad 
 |-  ( b e. RR* -> ( b =/= -oo -> -oo < b ) )
147 146 imp 
 |-  ( ( b e. RR* /\ b =/= -oo ) -> -oo < b )
148 143 147 sylan 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> -oo < b )
149 simpr 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> b e. RR* )
150 9 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> ( S.1 ` f ) e. RR* )
151 149 150 anim12i 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( b e. RR* /\ ( S.1 ` f ) e. RR* ) )
152 xrltle 
 |-  ( ( b e. RR* /\ ( S.1 ` f ) e. RR* ) -> ( b < ( S.1 ` f ) -> b <_ ( S.1 ` f ) ) )
153 152 imp 
 |-  ( ( ( b e. RR* /\ ( S.1 ` f ) e. RR* ) /\ b < ( S.1 ` f ) ) -> b <_ ( S.1 ` f ) )
154 151 153 sylan 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> b <_ ( S.1 ` f ) )
155 154 adantr 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> b <_ ( S.1 ` f ) )
156 xrre 
 |-  ( ( ( b e. RR* /\ ( S.1 ` f ) e. RR ) /\ ( -oo < b /\ b <_ ( S.1 ` f ) ) ) -> b e. RR )
157 140 142 148 155 156 syl22anc 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> b e. RR )
158 127 ad3antrrr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ 0 = ( S.1 ` g ) ) )
159 simplrl 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> b < ( S.1 ` f ) )
160 simplrl 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> f e. dom S.1 )
161 simpl 
 |-  ( ( f e. dom S.1 /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> f e. dom S.1 )
162 cnvimass 
 |-  ( `' f " ( ran f \ { 0 } ) ) C_ dom f
163 162 35 fssdm 
 |-  ( f e. dom S.1 -> ( `' f " ( ran f \ { 0 } ) ) C_ RR )
164 163 adantr 
 |-  ( ( f e. dom S.1 /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> ( `' f " ( ran f \ { 0 } ) ) C_ RR )
165 simpr 
 |-  ( ( f e. dom S.1 /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 )
166 fdm 
 |-  ( f : RR --> RR -> dom f = RR )
167 166 eqcomd 
 |-  ( f : RR --> RR -> RR = dom f )
168 ffun 
 |-  ( f : RR --> RR -> Fun f )
169 difpreima 
 |-  ( Fun f -> ( `' f " ( ran f \ { 0 } ) ) = ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) )
170 168 169 syl 
 |-  ( f : RR --> RR -> ( `' f " ( ran f \ { 0 } ) ) = ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) )
171 cnvimarndm 
 |-  ( `' f " ran f ) = dom f
172 171 difeq1i 
 |-  ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) = ( dom f \ ( `' f " { 0 } ) )
173 170 172 eqtrdi 
 |-  ( f : RR --> RR -> ( `' f " ( ran f \ { 0 } ) ) = ( dom f \ ( `' f " { 0 } ) ) )
174 167 173 difeq12d 
 |-  ( f : RR --> RR -> ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) = ( dom f \ ( dom f \ ( `' f " { 0 } ) ) ) )
175 cnvimass 
 |-  ( `' f " { 0 } ) C_ dom f
176 dfss4 
 |-  ( ( `' f " { 0 } ) C_ dom f <-> ( dom f \ ( dom f \ ( `' f " { 0 } ) ) ) = ( `' f " { 0 } ) )
177 175 176 mpbi 
 |-  ( dom f \ ( dom f \ ( `' f " { 0 } ) ) ) = ( `' f " { 0 } )
178 174 177 eqtrdi 
 |-  ( f : RR --> RR -> ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) = ( `' f " { 0 } ) )
179 178 eleq2d 
 |-  ( f : RR --> RR -> ( z e. ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) <-> z e. ( `' f " { 0 } ) ) )
180 ffn 
 |-  ( f : RR --> RR -> f Fn RR )
181 fniniseg 
 |-  ( f Fn RR -> ( z e. ( `' f " { 0 } ) <-> ( z e. RR /\ ( f ` z ) = 0 ) ) )
182 simpr 
 |-  ( ( z e. RR /\ ( f ` z ) = 0 ) -> ( f ` z ) = 0 )
183 181 182 biimtrdi 
 |-  ( f Fn RR -> ( z e. ( `' f " { 0 } ) -> ( f ` z ) = 0 ) )
184 180 183 syl 
 |-  ( f : RR --> RR -> ( z e. ( `' f " { 0 } ) -> ( f ` z ) = 0 ) )
185 179 184 sylbid 
 |-  ( f : RR --> RR -> ( z e. ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) -> ( f ` z ) = 0 ) )
186 35 185 syl 
 |-  ( f e. dom S.1 -> ( z e. ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) -> ( f ` z ) = 0 ) )
187 186 imp 
 |-  ( ( f e. dom S.1 /\ z e. ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) ) -> ( f ` z ) = 0 )
188 187 adantlr 
 |-  ( ( ( f e. dom S.1 /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) /\ z e. ( RR \ ( `' f " ( ran f \ { 0 } ) ) ) ) -> ( f ` z ) = 0 )
189 161 164 165 188 itg10a 
 |-  ( ( f e. dom S.1 /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> ( S.1 ` f ) = 0 )
190 160 189 sylan 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> ( S.1 ` f ) = 0 )
191 159 190 breqtrd 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> b < 0 )
192 158 191 137 syl2anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) = 0 ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
193 simprl 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> f e. dom S.1 )
194 simpr 
 |-  ( ( b < ( S.1 ` f ) /\ b e. RR ) -> b e. RR )
195 193 194 anim12i 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> ( f e. dom S.1 /\ b e. RR ) )
196 63 a1i 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> RR e. _V )
197 fvex 
 |-  ( f ` u ) e. _V
198 197 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( f ` u ) e. _V )
199 ovex 
 |-  ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. _V
200 199 105 ifex 
 |-  if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) e. _V
201 200 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) e. _V )
202 35 feqmptd 
 |-  ( f e. dom S.1 -> f = ( u e. RR |-> ( f ` u ) ) )
203 202 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f = ( u e. RR |-> ( f ` u ) ) )
204 eqidd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
205 196 198 201 203 204 offval2 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) = ( u e. RR |-> ( ( f ` u ) - if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) )
206 ovif2 
 |-  ( ( f ` u ) - if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` u ) - 0 ) )
207 171 166 eqtrid 
 |-  ( f : RR --> RR -> ( `' f " ran f ) = RR )
208 207 difeq1d 
 |-  ( f : RR --> RR -> ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) = ( RR \ ( `' f " { 0 } ) ) )
209 170 208 eqtrd 
 |-  ( f : RR --> RR -> ( `' f " ( ran f \ { 0 } ) ) = ( RR \ ( `' f " { 0 } ) ) )
210 209 eleq2d 
 |-  ( f : RR --> RR -> ( u e. ( `' f " ( ran f \ { 0 } ) ) <-> u e. ( RR \ ( `' f " { 0 } ) ) ) )
211 35 210 syl 
 |-  ( f e. dom S.1 -> ( u e. ( `' f " ( ran f \ { 0 } ) ) <-> u e. ( RR \ ( `' f " { 0 } ) ) ) )
212 211 ad3antrrr 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( u e. ( `' f " ( ran f \ { 0 } ) ) <-> u e. ( RR \ ( `' f " { 0 } ) ) ) )
213 simpr 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> u e. RR )
214 213 biantrurd 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( -. u e. ( `' f " { 0 } ) <-> ( u e. RR /\ -. u e. ( `' f " { 0 } ) ) ) )
215 eldif 
 |-  ( u e. ( RR \ ( `' f " { 0 } ) ) <-> ( u e. RR /\ -. u e. ( `' f " { 0 } ) ) )
216 214 215 bitr4di 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( -. u e. ( `' f " { 0 } ) <-> u e. ( RR \ ( `' f " { 0 } ) ) ) )
217 212 216 bitr4d 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( u e. ( `' f " ( ran f \ { 0 } ) ) <-> -. u e. ( `' f " { 0 } ) ) )
218 217 con2bid 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( u e. ( `' f " { 0 } ) <-> -. u e. ( `' f " ( ran f \ { 0 } ) ) ) )
219 fniniseg 
 |-  ( f Fn RR -> ( u e. ( `' f " { 0 } ) <-> ( u e. RR /\ ( f ` u ) = 0 ) ) )
220 35 180 219 3syl 
 |-  ( f e. dom S.1 -> ( u e. ( `' f " { 0 } ) <-> ( u e. RR /\ ( f ` u ) = 0 ) ) )
221 220 ad3antrrr 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( u e. ( `' f " { 0 } ) <-> ( u e. RR /\ ( f ` u ) = 0 ) ) )
222 218 221 bitr3d 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( -. u e. ( `' f " ( ran f \ { 0 } ) ) <-> ( u e. RR /\ ( f ` u ) = 0 ) ) )
223 oveq1 
 |-  ( ( f ` u ) = 0 -> ( ( f ` u ) - 0 ) = ( 0 - 0 ) )
224 0m0e0 
 |-  ( 0 - 0 ) = 0
225 223 224 eqtrdi 
 |-  ( ( f ` u ) = 0 -> ( ( f ` u ) - 0 ) = 0 )
226 225 adantl 
 |-  ( ( u e. RR /\ ( f ` u ) = 0 ) -> ( ( f ` u ) - 0 ) = 0 )
227 222 226 biimtrdi 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( -. u e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( f ` u ) - 0 ) = 0 ) )
228 227 imp 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) /\ -. u e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` u ) - 0 ) = 0 )
229 228 ifeq2da 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` u ) - 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
230 206 229 eqtrid 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( ( f ` u ) - if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
231 230 mpteq2dva 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> ( ( f ` u ) - if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
232 205 231 eqtrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
233 simpll 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f e. dom S.1 )
234 199 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. _V )
235 1ex 
 |-  1 e. _V
236 235 105 ifex 
 |-  if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) e. _V
237 236 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ u e. RR ) -> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) e. _V )
238 fconstmpt 
 |-  ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) = ( u e. RR |-> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
239 238 a1i 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) = ( u e. RR |-> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
240 eqidd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) )
241 196 234 237 239 240 offval2 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( u e. RR |-> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) )
242 ovif2 
 |-  ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 1 ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 0 ) )
243 resubcl 
 |-  ( ( ( S.1 ` f ) e. RR /\ b e. RR ) -> ( ( S.1 ` f ) - b ) e. RR )
244 8 243 sylan 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( S.1 ` f ) - b ) e. RR )
245 244 adantr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - b ) e. RR )
246 2re 
 |-  2 e. RR
247 i1fima 
 |-  ( f e. dom S.1 -> ( `' f " ( ran f \ { 0 } ) ) e. dom vol )
248 mblvol 
 |-  ( ( `' f " ( ran f \ { 0 } ) ) e. dom vol -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) = ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
249 247 248 syl 
 |-  ( f e. dom S.1 -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) = ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
250 neldifsn 
 |-  -. 0 e. ( ran f \ { 0 } )
251 i1fima2 
 |-  ( ( f e. dom S.1 /\ -. 0 e. ( ran f \ { 0 } ) ) -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
252 250 251 mpan2 
 |-  ( f e. dom S.1 -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
253 249 252 eqeltrrd 
 |-  ( f e. dom S.1 -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
254 remulcl 
 |-  ( ( 2 e. RR /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. RR )
255 246 253 254 sylancr 
 |-  ( f e. dom S.1 -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. RR )
256 255 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. RR )
257 2cnd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 2 e. CC )
258 253 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
259 258 recnd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. CC )
260 2ne0 
 |-  2 =/= 0
261 260 a1i 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 2 =/= 0 )
262 simpr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 )
263 257 259 261 262 mulne0d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) =/= 0 )
264 245 256 263 redivcld 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. RR )
265 264 recnd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. CC )
266 265 mulridd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 1 ) = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
267 265 mul01d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 0 ) = 0 )
268 266 267 ifeq12d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 1 ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) )
269 242 268 eqtrid 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) )
270 269 mpteq2dv 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
271 241 270 eqtrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
272 eqid 
 |-  ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) )
273 272 i1f1 
 |-  ( ( ( `' f " ( ran f \ { 0 } ) ) e. dom vol /\ ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
274 247 252 273 syl2anc 
 |-  ( f e. dom S.1 -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
275 274 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
276 275 264 i1fmulc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) e. dom S.1 )
277 271 276 eqeltrrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. dom S.1 )
278 i1fsub 
 |-  ( ( f e. dom S.1 /\ ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. dom S.1 ) -> ( f oF - ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 )
279 233 277 278 syl2anc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 )
280 232 279 eqeltrrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 )
281 iftrue 
 |-  ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
282 iftrue 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
283 282 breq2d 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
284 283 282 ifbieq1d 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
285 iftrue 
 |-  ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
286 284 285 sylan9eqr 
 |-  ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
287 281 286 eqtr4d 
 |-  ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) )
288 iffalse 
 |-  ( -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 )
289 ianor 
 |-  ( -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) <-> ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) \/ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) )
290 283 ifbid 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) )
291 iffalse 
 |-  ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
292 290 291 sylan9eqr 
 |-  ( ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
293 292 ex 
 |-  ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 ) )
294 iffalse 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 )
295 eqid 
 |-  0 = 0
296 eqeq1 
 |-  ( if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) -> ( if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 <-> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 ) )
297 eqeq1 
 |-  ( 0 = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) -> ( 0 = 0 <-> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 ) )
298 296 297 ifboth 
 |-  ( ( if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 /\ 0 = 0 ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
299 294 295 298 sylancl 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
300 293 299 pm2.61d1 
 |-  ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
301 300 299 jaoi 
 |-  ( ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) \/ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
302 289 301 sylbi 
 |-  ( -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) = 0 )
303 288 302 eqtr4d 
 |-  ( -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) )
304 287 303 pm2.61i 
 |-  if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 )
305 eleq1w 
 |-  ( u = z -> ( u e. ( `' f " ( ran f \ { 0 } ) ) <-> z e. ( `' f " ( ran f \ { 0 } ) ) ) )
306 fveq2 
 |-  ( u = z -> ( f ` u ) = ( f ` z ) )
307 306 oveq1d 
 |-  ( u = z -> ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
308 305 307 ifbieq1d 
 |-  ( u = z -> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
309 eqid 
 |-  ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) = ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
310 ovex 
 |-  ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) e. _V
311 310 105 ifex 
 |-  if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) e. _V
312 308 309 311 fvmpt 
 |-  ( z e. RR -> ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) = if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
313 312 breq2d 
 |-  ( z e. RR -> ( 0 <_ ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) <-> 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
314 313 312 ifbieq1d 
 |-  ( z e. RR -> if ( 0 <_ ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , 0 ) = if ( 0 <_ if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) , 0 ) )
315 304 314 eqtr4id 
 |-  ( z e. RR -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , 0 ) )
316 315 mpteq2ia 
 |-  ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) = ( z e. RR |-> if ( 0 <_ ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) , 0 ) )
317 316 i1fpos 
 |-  ( ( u e. RR |-> if ( u e. ( `' f " ( ran f \ { 0 } ) ) , ( ( f ` u ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 -> ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 )
318 280 317 syl 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 )
319 195 318 sylan 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 )
320 195 264 sylan 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. RR )
321 8 ad2antrl 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( S.1 ` f ) e. RR )
322 321 194 243 syl2an 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> ( ( S.1 ` f ) - b ) e. RR )
323 322 adantr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - b ) e. RR )
324 255 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. RR )
325 324 ad3antlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. RR )
326 simprl 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> b < ( S.1 ` f ) )
327 simprr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> b e. RR )
328 141 ad2antlr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> ( S.1 ` f ) e. RR )
329 327 328 posdifd 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> ( b < ( S.1 ` f ) <-> 0 < ( ( S.1 ` f ) - b ) ) )
330 326 329 mpbid 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> 0 < ( ( S.1 ` f ) - b ) )
331 330 adantr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( ( S.1 ` f ) - b ) )
332 253 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
333 332 ad3antlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR )
334 mblss 
 |-  ( ( `' f " ( ran f \ { 0 } ) ) e. dom vol -> ( `' f " ( ran f \ { 0 } ) ) C_ RR )
335 ovolge0 
 |-  ( ( `' f " ( ran f \ { 0 } ) ) C_ RR -> 0 <_ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
336 247 334 335 3syl 
 |-  ( f e. dom S.1 -> 0 <_ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
337 ltlen 
 |-  ( ( 0 e. RR /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR ) -> ( 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) <-> ( 0 <_ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) ) )
338 48 253 337 sylancr 
 |-  ( f e. dom S.1 -> ( 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) <-> ( 0 <_ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) ) )
339 338 biimprd 
 |-  ( f e. dom S.1 -> ( ( 0 <_ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
340 336 339 mpand 
 |-  ( f e. dom S.1 -> ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 -> 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
341 340 ad2antrl 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 -> 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
342 341 imp 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
343 342 adantlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
344 2pos 
 |-  0 < 2
345 mulgt0 
 |-  ( ( ( 2 e. RR /\ 0 < 2 ) /\ ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR /\ 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) -> 0 < ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
346 246 344 345 mpanl12 
 |-  ( ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR /\ 0 < ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) -> 0 < ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
347 333 343 346 syl2anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
348 323 325 331 347 divgt0d 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> 0 < ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
349 320 348 elrpd 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. RR+ )
350 simprl 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> f oR <_ F )
351 350 ad3antlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f oR <_ F )
352 ffn 
 |-  ( F : RR --> ( 0 [,] +oo ) -> F Fn RR )
353 35 180 syl 
 |-  ( f e. dom S.1 -> f Fn RR )
354 353 adantr 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> f Fn RR )
355 simpr 
 |-  ( ( F Fn RR /\ f Fn RR ) -> f Fn RR )
356 simpl 
 |-  ( ( F Fn RR /\ f Fn RR ) -> F Fn RR )
357 63 a1i 
 |-  ( ( F Fn RR /\ f Fn RR ) -> RR e. _V )
358 inidm 
 |-  ( RR i^i RR ) = RR
359 eqidd 
 |-  ( ( ( F Fn RR /\ f Fn RR ) /\ z e. RR ) -> ( f ` z ) = ( f ` z ) )
360 eqidd 
 |-  ( ( ( F Fn RR /\ f Fn RR ) /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
361 355 356 357 357 358 359 360 ofrfval 
 |-  ( ( F Fn RR /\ f Fn RR ) -> ( f oR <_ F <-> A. z e. RR ( f ` z ) <_ ( F ` z ) ) )
362 352 354 361 syl2an 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( f oR <_ F <-> A. z e. RR ( f ` z ) <_ ( F ` z ) ) )
363 362 ad2antrr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oR <_ F <-> A. z e. RR ( f ` z ) <_ ( F ` z ) ) )
364 simpl 
 |-  ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> f e. dom S.1 )
365 364 anim2i 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) )
366 365 194 anim12i 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) )
367 breq1 
 |-  ( 0 = if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( 0 <_ ( F ` z ) <-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
368 breq1 
 |-  ( ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) <-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
369 simplll 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> F : RR --> ( 0 [,] +oo ) )
370 369 ffvelcdmda 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( F ` z ) e. ( 0 [,] +oo ) )
371 370 100 sylib 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( F ` z ) e. RR* /\ 0 <_ ( F ` z ) ) )
372 371 simprd 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> 0 <_ ( F ` z ) )
373 372 ad2antrr 
 |-  ( ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) /\ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 ) -> 0 <_ ( F ` z ) )
374 oveq1 
 |-  ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
375 374 breq1d 
 |-  ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) -> ( ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) <-> ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) ) )
376 oveq1 
 |-  ( 0 = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) -> ( 0 + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
377 376 breq1d 
 |-  ( 0 = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) -> ( ( 0 + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) <-> ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) ) )
378 35 ad3antlr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f : RR --> RR )
379 378 ffvelcdmda 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( f ` z ) e. RR )
380 379 recnd 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( f ` z ) e. CC )
381 244 recnd 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( S.1 ` f ) - b ) e. CC )
382 381 adantr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - b ) e. CC )
383 255 recnd 
 |-  ( f e. dom S.1 -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. CC )
384 383 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) e. CC )
385 382 384 263 divcld 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. CC )
386 385 adantlll 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. CC )
387 386 adantr 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. CC )
388 380 387 npcand 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( f ` z ) )
389 388 adantr 
 |-  ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( f ` z ) )
390 simpr 
 |-  ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) -> ( f ` z ) <_ ( F ` z ) )
391 389 390 eqbrtrd 
 |-  ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) )
392 391 ad2antrr 
 |-  ( ( ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) /\ -. if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 ) /\ ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) )
393 288 pm2.24d 
 |-  ( -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( -. if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 -> ( 0 + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) ) )
394 393 impcom 
 |-  ( ( -. if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 /\ -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) ) -> ( 0 + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) )
395 394 adantll 
 |-  ( ( ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) /\ -. if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 ) /\ -. ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) ) -> ( 0 + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) )
396 375 377 392 395 ifbothda 
 |-  ( ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) /\ -. if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 ) -> ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( F ` z ) )
397 367 368 373 396 ifbothda 
 |-  ( ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ ( f ` z ) <_ ( F ` z ) ) -> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) )
398 397 ex 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ f e. dom S.1 ) /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) <_ ( F ` z ) -> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
399 366 398 sylanl1 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) <_ ( F ` z ) -> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
400 399 ralimdva 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( A. z e. RR ( f ` z ) <_ ( F ` z ) -> A. z e. RR if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
401 363 400 sylbid 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oR <_ F -> A. z e. RR if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
402 351 401 mpd 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> A. z e. RR if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) )
403 ovex 
 |-  ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) e. _V
404 105 403 ifex 
 |-  if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) e. _V
405 404 a1i 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ z e. RR ) -> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) e. _V )
406 eqidd 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) = ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) )
407 104 405 99 406 67 ofrfval2 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) oR <_ F <-> A. z e. RR if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
408 407 ad3antrrr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) oR <_ F <-> A. z e. RR if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) <_ ( F ` z ) ) )
409 402 408 mpbird 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) oR <_ F )
410 oveq2 
 |-  ( y = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) -> ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) = ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
411 410 ifeq2d 
 |-  ( y = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) -> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) = if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
412 411 mpteq2dv 
 |-  ( y = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) -> ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) = ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) )
413 412 breq1d 
 |-  ( y = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) -> ( ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F <-> ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) oR <_ F ) )
414 413 rspcev 
 |-  ( ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. RR+ /\ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) ) oR <_ F ) -> E. y e. RR+ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F )
415 349 409 414 syl2anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> E. y e. RR+ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F )
416 fveq2 
 |-  ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) = g -> ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) )
417 416 eqcoms 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) )
418 417 biantrud 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) ) )
419 nfmpt1 
 |-  F/_ z ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
420 419 nfeq2 
 |-  F/ z g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
421 fveq1 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( g ` z ) = ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) )
422 310 105 ifex 
 |-  if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) e. _V
423 eqid 
 |-  ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
424 423 fvmpt2 
 |-  ( ( z e. RR /\ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) e. _V ) -> ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
425 422 424 mpan2 
 |-  ( z e. RR -> ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ` z ) = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
426 421 425 sylan9eq 
 |-  ( ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) /\ z e. RR ) -> ( g ` z ) = if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
427 426 eqeq1d 
 |-  ( ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) /\ z e. RR ) -> ( ( g ` z ) = 0 <-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 ) )
428 426 oveq1d 
 |-  ( ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) /\ z e. RR ) -> ( ( g ` z ) + y ) = ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) )
429 427 428 ifbieq2d 
 |-  ( ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) /\ z e. RR ) -> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) = if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) )
430 420 429 mpteq2da 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) = ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) )
431 430 breq1d 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F ) )
432 431 rexbidv 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F <-> E. y e. RR+ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F ) )
433 418 432 bitr3d 
 |-  ( g = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) <-> E. y e. RR+ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F ) )
434 433 rspcev 
 |-  ( ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 /\ E. y e. RR+ ( z e. RR |-> if ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 , 0 , ( if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) + y ) ) ) oR <_ F ) -> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) )
435 319 415 434 syl2anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) )
436 simplrr 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b e. RR )
437 199 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. _V )
438 235 105 ifex 
 |-  if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) e. _V
439 438 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) e. _V )
440 fconstmpt 
 |-  ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) = ( z e. RR |-> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
441 440 a1i 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) = ( z e. RR |-> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
442 eqidd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) )
443 196 437 439 441 442 offval2 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( z e. RR |-> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) )
444 ovif2 
 |-  ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 1 ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 0 ) )
445 266 267 ifeq12d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 1 ) , ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. 0 ) ) = if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) )
446 444 445 eqtrid 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) )
447 446 mpteq2dv 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
448 443 447 eqtrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
449 eqid 
 |-  ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) = ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) )
450 449 i1f1 
 |-  ( ( ( `' f " ( ran f \ { 0 } ) ) e. dom vol /\ ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR ) -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
451 247 252 450 syl2anc 
 |-  ( f e. dom S.1 -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
452 451 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) e. dom S.1 )
453 452 264 i1fmulc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) e. dom S.1 )
454 448 453 eqeltrrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. dom S.1 )
455 i1fsub 
 |-  ( ( f e. dom S.1 /\ ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. dom S.1 ) -> ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 )
456 233 454 455 syl2anc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 )
457 itg1cl 
 |-  ( ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) e. RR )
458 456 457 syl 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) e. RR )
459 458 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) e. RR )
460 318 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 )
461 itg1cl 
 |-  ( ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 -> ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) e. RR )
462 460 461 syl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) e. RR )
463 simplrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( S.1 ` f ) )
464 simpr 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> b e. RR )
465 8 adantr 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( S.1 ` f ) e. RR )
466 97 a1i 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> 2 e. RR+ )
467 464 465 466 ltdiv1d 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( b < ( S.1 ` f ) <-> ( b / 2 ) < ( ( S.1 ` f ) / 2 ) ) )
468 recn 
 |-  ( b e. RR -> b e. CC )
469 468 2halvesd 
 |-  ( b e. RR -> ( ( b / 2 ) + ( b / 2 ) ) = b )
470 469 oveq1d 
 |-  ( b e. RR -> ( ( ( b / 2 ) + ( b / 2 ) ) - ( b / 2 ) ) = ( b - ( b / 2 ) ) )
471 468 halfcld 
 |-  ( b e. RR -> ( b / 2 ) e. CC )
472 471 471 pncand 
 |-  ( b e. RR -> ( ( ( b / 2 ) + ( b / 2 ) ) - ( b / 2 ) ) = ( b / 2 ) )
473 470 472 eqtr3d 
 |-  ( b e. RR -> ( b - ( b / 2 ) ) = ( b / 2 ) )
474 473 breq1d 
 |-  ( b e. RR -> ( ( b - ( b / 2 ) ) < ( ( S.1 ` f ) / 2 ) <-> ( b / 2 ) < ( ( S.1 ` f ) / 2 ) ) )
475 474 adantl 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( b - ( b / 2 ) ) < ( ( S.1 ` f ) / 2 ) <-> ( b / 2 ) < ( ( S.1 ` f ) / 2 ) ) )
476 rehalfcl 
 |-  ( b e. RR -> ( b / 2 ) e. RR )
477 476 adantl 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( b / 2 ) e. RR )
478 8 rehalfcld 
 |-  ( f e. dom S.1 -> ( ( S.1 ` f ) / 2 ) e. RR )
479 478 adantr 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( S.1 ` f ) / 2 ) e. RR )
480 464 477 479 ltsubaddd 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( b - ( b / 2 ) ) < ( ( S.1 ` f ) / 2 ) <-> b < ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) ) )
481 467 475 480 3bitr2d 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( b < ( S.1 ` f ) <-> b < ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) ) )
482 481 adantr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( b < ( S.1 ` f ) <-> b < ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) ) )
483 482 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( b < ( S.1 ` f ) <-> b < ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) ) )
484 463 483 mpbid 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) )
485 452 264 itg1mulc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) = ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) )
486 448 fveq2d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( ( RR X. { ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) } ) oF x. ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) = ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) )
487 449 itg11 
 |-  ( ( ( `' f " ( ran f \ { 0 } ) ) e. dom vol /\ ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. RR ) -> ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) )
488 247 252 487 syl2anc 
 |-  ( f e. dom S.1 -> ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) = ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) )
489 488 oveq2d 
 |-  ( f e. dom S.1 -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) = ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
490 489 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) = ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
491 252 recnd 
 |-  ( f e. dom S.1 -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. CC )
492 491 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) e. CC )
493 265 492 mulcomd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) ) = ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
494 249 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) = ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) )
495 494 oveq1d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) = ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) )
496 259 382 mulcomd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) = ( ( ( S.1 ` f ) - b ) x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
497 495 496 eqtrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) = ( ( ( S.1 ` f ) - b ) x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) )
498 497 oveq1d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) = ( ( ( ( S.1 ` f ) - b ) x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
499 492 382 384 263 divassd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( S.1 ` f ) - b ) ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) = ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
500 382 257 259 261 262 divcan5rd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) = ( ( ( S.1 ` f ) - b ) / 2 ) )
501 498 499 500 3eqtr3d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( vol ` ( `' f " ( ran f \ { 0 } ) ) ) x. ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) = ( ( ( S.1 ` f ) - b ) / 2 ) )
502 490 493 501 3eqtrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) x. ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , 1 , 0 ) ) ) ) = ( ( ( S.1 ` f ) - b ) / 2 ) )
503 485 486 502 3eqtr3d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) = ( ( ( S.1 ` f ) - b ) / 2 ) )
504 503 oveq2d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) = ( ( S.1 ` f ) - ( ( ( S.1 ` f ) - b ) / 2 ) ) )
505 itg1sub 
 |-  ( ( f e. dom S.1 /\ ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. dom S.1 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) = ( ( S.1 ` f ) - ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) )
506 233 454 505 syl2anc 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) = ( ( S.1 ` f ) - ( S.1 ` ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) )
507 8 recnd 
 |-  ( f e. dom S.1 -> ( S.1 ` f ) e. CC )
508 507 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` f ) e. CC )
509 468 ad2antlr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b e. CC )
510 508 509 257 261 divsubdird 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - b ) / 2 ) = ( ( ( S.1 ` f ) / 2 ) - ( b / 2 ) ) )
511 510 oveq2d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - ( ( ( S.1 ` f ) - b ) / 2 ) ) = ( ( S.1 ` f ) - ( ( ( S.1 ` f ) / 2 ) - ( b / 2 ) ) ) )
512 507 adantr 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( S.1 ` f ) e. CC )
513 512 halfcld 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( S.1 ` f ) / 2 ) e. CC )
514 471 adantl 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( b / 2 ) e. CC )
515 512 513 514 subsubd 
 |-  ( ( f e. dom S.1 /\ b e. RR ) -> ( ( S.1 ` f ) - ( ( ( S.1 ` f ) / 2 ) - ( b / 2 ) ) ) = ( ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) + ( b / 2 ) ) )
516 515 adantr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( S.1 ` f ) - ( ( ( S.1 ` f ) / 2 ) - ( b / 2 ) ) ) = ( ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) + ( b / 2 ) ) )
517 507 2halvesd 
 |-  ( f e. dom S.1 -> ( ( ( S.1 ` f ) / 2 ) + ( ( S.1 ` f ) / 2 ) ) = ( S.1 ` f ) )
518 517 oveq1d 
 |-  ( f e. dom S.1 -> ( ( ( ( S.1 ` f ) / 2 ) + ( ( S.1 ` f ) / 2 ) ) - ( ( S.1 ` f ) / 2 ) ) = ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) )
519 507 halfcld 
 |-  ( f e. dom S.1 -> ( ( S.1 ` f ) / 2 ) e. CC )
520 519 519 pncand 
 |-  ( f e. dom S.1 -> ( ( ( ( S.1 ` f ) / 2 ) + ( ( S.1 ` f ) / 2 ) ) - ( ( S.1 ` f ) / 2 ) ) = ( ( S.1 ` f ) / 2 ) )
521 518 520 eqtr3d 
 |-  ( f e. dom S.1 -> ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) = ( ( S.1 ` f ) / 2 ) )
522 521 oveq1d 
 |-  ( f e. dom S.1 -> ( ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) + ( b / 2 ) ) = ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) )
523 522 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) - ( ( S.1 ` f ) / 2 ) ) + ( b / 2 ) ) = ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) )
524 511 516 523 3eqtrrd 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) = ( ( S.1 ` f ) - ( ( ( S.1 ` f ) - b ) / 2 ) ) )
525 504 506 524 3eqtr4d 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) = ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) )
526 525 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) = ( ( ( S.1 ` f ) / 2 ) + ( b / 2 ) ) )
527 484 526 breqtrrd 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) )
528 456 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 )
529 id 
 |-  ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) )
530 529 adantlrl 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) )
531 233 36 sylan 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( f ` z ) e. RR )
532 264 adantr 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) e. RR )
533 531 532 resubcld 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) e. RR )
534 533 leidd 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
535 534 adantr 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
536 285 breq2d 
 |-  ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
537 536 adantl 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
538 535 537 mpbird 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
539 533 adantr 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) e. RR )
540 48 a1i 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> 0 e. RR )
541 48 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> 0 e. RR )
542 533 541 ltnled 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) < 0 <-> -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
543 542 biimpar 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) < 0 )
544 539 540 543 ltled 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ 0 )
545 iffalse 
 |-  ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 )
546 545 breq2d 
 |-  ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ 0 ) )
547 546 adantl 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ 0 ) )
548 544 547 mpbird 
 |-  ( ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
549 538 548 pm2.61dan 
 |-  ( ( ( ( f e. dom S.1 /\ b e. RR ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
550 530 549 sylan 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
551 550 adantr 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
552 iftrue 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) = ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) )
553 552 oveq2d 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) )
554 iba 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <-> ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) ) )
555 554 bicomd 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) <-> 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) ) )
556 555 ifbid 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
557 553 556 breq12d 
 |-  ( z e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
558 557 adantl 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) <_ if ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
559 551 558 mpbird 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
560 35 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f : RR --> RR )
561 170 eleq2d 
 |-  ( f : RR --> RR -> ( z e. ( `' f " ( ran f \ { 0 } ) ) <-> z e. ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) ) )
562 eldif 
 |-  ( z e. ( ( `' f " ran f ) \ ( `' f " { 0 } ) ) <-> ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) )
563 561 562 bitrdi 
 |-  ( f : RR --> RR -> ( z e. ( `' f " ( ran f \ { 0 } ) ) <-> ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) ) )
564 563 notbid 
 |-  ( f : RR --> RR -> ( -. z e. ( `' f " ( ran f \ { 0 } ) ) <-> -. ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) ) )
565 564 adantr 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( -. z e. ( `' f " ( ran f \ { 0 } ) ) <-> -. ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) ) )
566 pm4.53 
 |-  ( -. ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) <-> ( -. z e. ( `' f " ran f ) \/ z e. ( `' f " { 0 } ) ) )
567 207 eleq2d 
 |-  ( f : RR --> RR -> ( z e. ( `' f " ran f ) <-> z e. RR ) )
568 567 biimpar 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> z e. ( `' f " ran f ) )
569 568 pm2.24d 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( -. z e. ( `' f " ran f ) -> ( f ` z ) = 0 ) )
570 181 simplbda 
 |-  ( ( f Fn RR /\ z e. ( `' f " { 0 } ) ) -> ( f ` z ) = 0 )
571 570 ex 
 |-  ( f Fn RR -> ( z e. ( `' f " { 0 } ) -> ( f ` z ) = 0 ) )
572 180 571 syl 
 |-  ( f : RR --> RR -> ( z e. ( `' f " { 0 } ) -> ( f ` z ) = 0 ) )
573 572 adantr 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( z e. ( `' f " { 0 } ) -> ( f ` z ) = 0 ) )
574 569 573 jaod 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( ( -. z e. ( `' f " ran f ) \/ z e. ( `' f " { 0 } ) ) -> ( f ` z ) = 0 ) )
575 566 574 biimtrid 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( -. ( z e. ( `' f " ran f ) /\ -. z e. ( `' f " { 0 } ) ) -> ( f ` z ) = 0 ) )
576 565 575 sylbid 
 |-  ( ( f : RR --> RR /\ z e. RR ) -> ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> ( f ` z ) = 0 ) )
577 576 imp 
 |-  ( ( ( f : RR --> RR /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( f ` z ) = 0 )
578 560 577 sylanl1 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( f ` z ) = 0 )
579 578 oveq1d 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - 0 ) = ( 0 - 0 ) )
580 579 224 eqtrdi 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - 0 ) = 0 )
581 580 30 eqbrtrdi 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - 0 ) <_ 0 )
582 iffalse 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) = 0 )
583 582 oveq2d 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = ( ( f ` z ) - 0 ) )
584 289 288 sylbir 
 |-  ( ( -. 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) \/ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 )
585 584 olcs 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) = 0 )
586 583 585 breq12d 
 |-  ( -. z e. ( `' f " ( ran f \ { 0 } ) ) -> ( ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - 0 ) <_ 0 ) )
587 586 adantl 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) <-> ( ( f ` z ) - 0 ) <_ 0 ) )
588 581 587 mpbird 
 |-  ( ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) /\ -. z e. ( `' f " ( ran f \ { 0 } ) ) ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
589 559 588 pm2.61dan 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
590 589 ralrimiva 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> A. z e. RR ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) )
591 63 a1i 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> RR e. _V )
592 ovex 
 |-  ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. _V
593 592 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) e. _V )
594 422 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) e. _V )
595 fvex 
 |-  ( f ` z ) e. _V
596 595 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> ( f ` z ) e. _V )
597 199 105 ifex 
 |-  if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) e. _V
598 597 a1i 
 |-  ( ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) /\ z e. RR ) -> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) e. _V )
599 70 ad2antrr 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> f = ( z e. RR |-> ( f ` z ) ) )
600 eqidd 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) = ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) )
601 591 596 598 599 600 offval2 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) = ( z e. RR |-> ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) )
602 eqidd 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) = ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
603 591 593 594 601 602 ofrfval2 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) oR <_ ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) <-> A. z e. RR ( ( f ` z ) - if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) <_ if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
604 590 603 mpbird 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) oR <_ ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) )
605 itg1le 
 |-  ( ( ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) e. dom S.1 /\ ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) e. dom S.1 /\ ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) oR <_ ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) <_ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) )
606 528 460 604 605 syl3anc 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> ( S.1 ` ( f oF - ( z e. RR |-> if ( z e. ( `' f " ( ran f \ { 0 } ) ) , ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) , 0 ) ) ) ) <_ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) )
607 436 459 462 527 606 ltletrd 
 |-  ( ( ( f e. dom S.1 /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) )
608 607 adantllr 
 |-  ( ( ( ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) )
609 608 adantlll 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) )
610 fvex 
 |-  ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) e. _V
611 eqeq1 
 |-  ( a = ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( a = ( S.1 ` g ) <-> ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) )
612 611 anbi2d 
 |-  ( a = ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) <-> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) ) )
613 612 rexbidv 
 |-  ( a = ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) <-> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) ) )
614 breq2 
 |-  ( a = ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( b < a <-> b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) ) )
615 613 614 anbi12d 
 |-  ( a = ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) -> ( ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) <-> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) /\ b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) ) ) )
616 610 615 spcev 
 |-  ( ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) = ( S.1 ` g ) ) /\ b < ( S.1 ` ( z e. RR |-> if ( ( 0 <_ ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) /\ z e. ( `' f " ( ran f \ { 0 } ) ) ) , ( ( f ` z ) - ( ( ( S.1 ` f ) - b ) / ( 2 x. ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) ) ) ) , 0 ) ) ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
617 435 609 616 syl2anc 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) /\ ( vol* ` ( `' f " ( ran f \ { 0 } ) ) ) =/= 0 ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
618 192 617 pm2.61dane 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ ( b < ( S.1 ` f ) /\ b e. RR ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
619 618 expr 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> ( b e. RR -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
620 619 adantllr 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> ( b e. RR -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
621 620 adantr 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> ( b e. RR -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
622 157 621 mpd 
 |-  ( ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) /\ b =/= -oo ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
623 139 622 pm2.61dane 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < ( S.1 ` f ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
624 623 ex 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( b < ( S.1 ` f ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
625 94 624 sylbid 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> ( b < s -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
626 625 imp 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) /\ b < s ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
627 626 an32s 
 |-  ( ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ b < s ) /\ ( f e. dom S.1 /\ ( f oR <_ F /\ s = ( S.1 ` f ) ) ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
628 627 rexlimdvaa 
 |-  ( ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) /\ b < s ) -> ( E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
629 628 expimpd 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( ( b < s /\ E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
630 629 ancomsd 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( ( E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) /\ b < s ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
631 630 exlimdv 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( E. s ( E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) /\ b < s ) -> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) ) )
632 eqeq1 
 |-  ( x = s -> ( x = ( S.1 ` f ) <-> s = ( S.1 ` f ) ) )
633 632 anbi2d 
 |-  ( x = s -> ( ( f oR <_ F /\ x = ( S.1 ` f ) ) <-> ( f oR <_ F /\ s = ( S.1 ` f ) ) ) )
634 633 rexbidv 
 |-  ( x = s -> ( E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) <-> E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) ) )
635 634 rexab 
 |-  ( E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s <-> E. s ( E. f e. dom S.1 ( f oR <_ F /\ s = ( S.1 ` f ) ) /\ b < s ) )
636 eqeq1 
 |-  ( x = a -> ( x = ( S.1 ` g ) <-> a = ( S.1 ` g ) ) )
637 636 anbi2d 
 |-  ( x = a -> ( ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) <-> ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) ) )
638 637 rexbidv 
 |-  ( x = a -> ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) <-> E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) ) )
639 638 rexab 
 |-  ( E. a e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } b < a <-> E. a ( E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ a = ( S.1 ` g ) ) /\ b < a ) )
640 631 635 639 3imtr4g 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( E. s e. { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } b < s -> E. a e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } b < a ) )
641 92 640 sylbid 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ b e. RR* ) -> ( b < sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) -> E. a e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } b < a ) )
642 641 impr 
 |-  ( ( F : RR --> ( 0 [,] +oo ) /\ ( b e. RR* /\ b < sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) ) ) -> E. a e. { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } b < a )
643 6 15 89 642 eqsupd 
 |-  ( F : RR --> ( 0 [,] +oo ) -> sup ( { x | E. g e. dom S.1 ( E. y e. RR+ ( z e. RR |-> if ( ( g ` z ) = 0 , 0 , ( ( g ` z ) + y ) ) ) oR <_ F /\ x = ( S.1 ` g ) ) } , RR* , < ) = sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) )
644 4 643 eqtrid 
 |-  ( F : RR --> ( 0 [,] +oo ) -> sup ( L , RR* , < ) = sup ( { x | E. f e. dom S.1 ( f oR <_ F /\ x = ( S.1 ` f ) ) } , RR* , < ) )
645 3 644 eqtr4d 
 |-  ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) = sup ( L , RR* , < ) )