Step |
Hyp |
Ref |
Expression |
1 |
|
ovolval4lem1.f |
|- ( ph -> F : NN --> ( RR* X. RR* ) ) |
2 |
|
ovolval4lem1.g |
|- G = ( n e. NN |-> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. ) |
3 |
|
ovolval4lem1.a |
|- A = { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } |
4 |
|
ioof |
|- (,) : ( RR* X. RR* ) --> ~P RR |
5 |
4
|
a1i |
|- ( ph -> (,) : ( RR* X. RR* ) --> ~P RR ) |
6 |
|
fco |
|- ( ( (,) : ( RR* X. RR* ) --> ~P RR /\ F : NN --> ( RR* X. RR* ) ) -> ( (,) o. F ) : NN --> ~P RR ) |
7 |
5 1 6
|
syl2anc |
|- ( ph -> ( (,) o. F ) : NN --> ~P RR ) |
8 |
7
|
ffnd |
|- ( ph -> ( (,) o. F ) Fn NN ) |
9 |
|
fniunfv |
|- ( ( (,) o. F ) Fn NN -> U_ n e. NN ( ( (,) o. F ) ` n ) = U. ran ( (,) o. F ) ) |
10 |
8 9
|
syl |
|- ( ph -> U_ n e. NN ( ( (,) o. F ) ` n ) = U. ran ( (,) o. F ) ) |
11 |
10
|
eqcomd |
|- ( ph -> U. ran ( (,) o. F ) = U_ n e. NN ( ( (,) o. F ) ` n ) ) |
12 |
|
ssrab2 |
|- { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } C_ NN |
13 |
3 12
|
eqsstri |
|- A C_ NN |
14 |
|
undif |
|- ( A C_ NN <-> ( A u. ( NN \ A ) ) = NN ) |
15 |
13 14
|
mpbi |
|- ( A u. ( NN \ A ) ) = NN |
16 |
15
|
eqcomi |
|- NN = ( A u. ( NN \ A ) ) |
17 |
16
|
iuneq1i |
|- U_ n e. NN ( ( (,) o. F ) ` n ) = U_ n e. ( A u. ( NN \ A ) ) ( ( (,) o. F ) ` n ) |
18 |
|
iunxun |
|- U_ n e. ( A u. ( NN \ A ) ) ( ( (,) o. F ) ` n ) = ( U_ n e. A ( ( (,) o. F ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) |
19 |
17 18
|
eqtri |
|- U_ n e. NN ( ( (,) o. F ) ` n ) = ( U_ n e. A ( ( (,) o. F ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) |
20 |
19
|
a1i |
|- ( ph -> U_ n e. NN ( ( (,) o. F ) ` n ) = ( U_ n e. A ( ( (,) o. F ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) ) |
21 |
1
|
ffvelrnda |
|- ( ( ph /\ n e. NN ) -> ( F ` n ) e. ( RR* X. RR* ) ) |
22 |
|
xp1st |
|- ( ( F ` n ) e. ( RR* X. RR* ) -> ( 1st ` ( F ` n ) ) e. RR* ) |
23 |
21 22
|
syl |
|- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR* ) |
24 |
|
xp2nd |
|- ( ( F ` n ) e. ( RR* X. RR* ) -> ( 2nd ` ( F ` n ) ) e. RR* ) |
25 |
21 24
|
syl |
|- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR* ) |
26 |
25 23
|
ifcld |
|- ( ( ph /\ n e. NN ) -> if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) e. RR* ) |
27 |
23 26
|
opelxpd |
|- ( ( ph /\ n e. NN ) -> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. e. ( RR* X. RR* ) ) |
28 |
27 2
|
fmptd |
|- ( ph -> G : NN --> ( RR* X. RR* ) ) |
29 |
|
fco |
|- ( ( (,) : ( RR* X. RR* ) --> ~P RR /\ G : NN --> ( RR* X. RR* ) ) -> ( (,) o. G ) : NN --> ~P RR ) |
30 |
5 28 29
|
syl2anc |
|- ( ph -> ( (,) o. G ) : NN --> ~P RR ) |
31 |
30
|
ffnd |
|- ( ph -> ( (,) o. G ) Fn NN ) |
32 |
|
fniunfv |
|- ( ( (,) o. G ) Fn NN -> U_ n e. NN ( ( (,) o. G ) ` n ) = U. ran ( (,) o. G ) ) |
33 |
31 32
|
syl |
|- ( ph -> U_ n e. NN ( ( (,) o. G ) ` n ) = U. ran ( (,) o. G ) ) |
34 |
33
|
eqcomd |
|- ( ph -> U. ran ( (,) o. G ) = U_ n e. NN ( ( (,) o. G ) ` n ) ) |
35 |
16
|
iuneq1i |
|- U_ n e. NN ( ( (,) o. G ) ` n ) = U_ n e. ( A u. ( NN \ A ) ) ( ( (,) o. G ) ` n ) |
36 |
|
iunxun |
|- U_ n e. ( A u. ( NN \ A ) ) ( ( (,) o. G ) ` n ) = ( U_ n e. A ( ( (,) o. G ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) ) |
37 |
35 36
|
eqtri |
|- U_ n e. NN ( ( (,) o. G ) ` n ) = ( U_ n e. A ( ( (,) o. G ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) ) |
38 |
37
|
a1i |
|- ( ph -> U_ n e. NN ( ( (,) o. G ) ` n ) = ( U_ n e. A ( ( (,) o. G ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) ) ) |
39 |
28
|
adantr |
|- ( ( ph /\ n e. A ) -> G : NN --> ( RR* X. RR* ) ) |
40 |
13
|
sseli |
|- ( n e. A -> n e. NN ) |
41 |
40
|
adantl |
|- ( ( ph /\ n e. A ) -> n e. NN ) |
42 |
|
fvco3 |
|- ( ( G : NN --> ( RR* X. RR* ) /\ n e. NN ) -> ( ( (,) o. G ) ` n ) = ( (,) ` ( G ` n ) ) ) |
43 |
39 41 42
|
syl2anc |
|- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) ` n ) = ( (,) ` ( G ` n ) ) ) |
44 |
1
|
adantr |
|- ( ( ph /\ n e. A ) -> F : NN --> ( RR* X. RR* ) ) |
45 |
|
fvco3 |
|- ( ( F : NN --> ( RR* X. RR* ) /\ n e. NN ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) ) |
46 |
44 41 45
|
syl2anc |
|- ( ( ph /\ n e. A ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) ) |
47 |
|
simpl |
|- ( ( ph /\ n e. A ) -> ph ) |
48 |
|
1st2nd2 |
|- ( ( F ` n ) e. ( RR* X. RR* ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
49 |
21 48
|
syl |
|- ( ( ph /\ n e. NN ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
50 |
47 41 49
|
syl2anc |
|- ( ( ph /\ n e. A ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
51 |
2
|
a1i |
|- ( ph -> G = ( n e. NN |-> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. ) ) |
52 |
27
|
elexd |
|- ( ( ph /\ n e. NN ) -> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. e. _V ) |
53 |
51 52
|
fvmpt2d |
|- ( ( ph /\ n e. NN ) -> ( G ` n ) = <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. ) |
54 |
47 41 53
|
syl2anc |
|- ( ( ph /\ n e. A ) -> ( G ` n ) = <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. ) |
55 |
3
|
eleq2i |
|- ( n e. A <-> n e. { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } ) |
56 |
55
|
biimpi |
|- ( n e. A -> n e. { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } ) |
57 |
|
rabid |
|- ( n e. { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } <-> ( n e. NN /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) ) |
58 |
56 57
|
sylib |
|- ( n e. A -> ( n e. NN /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) ) |
59 |
58
|
simprd |
|- ( n e. A -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) |
60 |
59
|
adantl |
|- ( ( ph /\ n e. A ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) |
61 |
60
|
iftrued |
|- ( ( ph /\ n e. A ) -> if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) = ( 2nd ` ( F ` n ) ) ) |
62 |
61
|
opeq2d |
|- ( ( ph /\ n e. A ) -> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
63 |
|
eqidd |
|- ( ( ph /\ n e. A ) -> <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
64 |
54 62 63
|
3eqtrd |
|- ( ( ph /\ n e. A ) -> ( G ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
65 |
50 64
|
eqtr4d |
|- ( ( ph /\ n e. A ) -> ( F ` n ) = ( G ` n ) ) |
66 |
65
|
fveq2d |
|- ( ( ph /\ n e. A ) -> ( (,) ` ( F ` n ) ) = ( (,) ` ( G ` n ) ) ) |
67 |
46 66
|
eqtrd |
|- ( ( ph /\ n e. A ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( G ` n ) ) ) |
68 |
43 67
|
eqtr4d |
|- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) ` n ) = ( ( (,) o. F ) ` n ) ) |
69 |
68
|
iuneq2dv |
|- ( ph -> U_ n e. A ( ( (,) o. G ) ` n ) = U_ n e. A ( ( (,) o. F ) ` n ) ) |
70 |
28
|
adantr |
|- ( ( ph /\ n e. ( NN \ A ) ) -> G : NN --> ( RR* X. RR* ) ) |
71 |
|
eldifi |
|- ( n e. ( NN \ A ) -> n e. NN ) |
72 |
71
|
adantl |
|- ( ( ph /\ n e. ( NN \ A ) ) -> n e. NN ) |
73 |
70 72 42
|
syl2anc |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. G ) ` n ) = ( (,) ` ( G ` n ) ) ) |
74 |
|
simpl |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ph ) |
75 |
74 72 53
|
syl2anc |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( G ` n ) = <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. ) |
76 |
71
|
anim1i |
|- ( ( n e. ( NN \ A ) /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> ( n e. NN /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) ) |
77 |
76 57
|
sylibr |
|- ( ( n e. ( NN \ A ) /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> n e. { n e. NN | ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) } ) |
78 |
77 55
|
sylibr |
|- ( ( n e. ( NN \ A ) /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> n e. A ) |
79 |
78
|
adantll |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> n e. A ) |
80 |
|
eldifn |
|- ( n e. ( NN \ A ) -> -. n e. A ) |
81 |
80
|
ad2antlr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> -. n e. A ) |
82 |
79 81
|
pm2.65da |
|- ( ( ph /\ n e. ( NN \ A ) ) -> -. ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) |
83 |
82
|
iffalsed |
|- ( ( ph /\ n e. ( NN \ A ) ) -> if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) = ( 1st ` ( F ` n ) ) ) |
84 |
83
|
opeq2d |
|- ( ( ph /\ n e. ( NN \ A ) ) -> <. ( 1st ` ( F ` n ) ) , if ( ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) ) >. = <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) |
85 |
75 84
|
eqtrd |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( G ` n ) = <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) |
86 |
85
|
fveq2d |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( (,) ` ( G ` n ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) ) |
87 |
|
iooid |
|- ( ( 1st ` ( F ` n ) ) (,) ( 1st ` ( F ` n ) ) ) = (/) |
88 |
87
|
eqcomi |
|- (/) = ( ( 1st ` ( F ` n ) ) (,) ( 1st ` ( F ` n ) ) ) |
89 |
|
df-ov |
|- ( ( 1st ` ( F ` n ) ) (,) ( 1st ` ( F ` n ) ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) |
90 |
88 89
|
eqtr2i |
|- ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) = (/) |
91 |
90
|
a1i |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 1st ` ( F ` n ) ) >. ) = (/) ) |
92 |
73 86 91
|
3eqtrd |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. G ) ` n ) = (/) ) |
93 |
92
|
iuneq2dv |
|- ( ph -> U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) = U_ n e. ( NN \ A ) (/) ) |
94 |
|
iun0 |
|- U_ n e. ( NN \ A ) (/) = (/) |
95 |
94
|
a1i |
|- ( ph -> U_ n e. ( NN \ A ) (/) = (/) ) |
96 |
93 95
|
eqtrd |
|- ( ph -> U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) = (/) ) |
97 |
74 1
|
syl |
|- ( ( ph /\ n e. ( NN \ A ) ) -> F : NN --> ( RR* X. RR* ) ) |
98 |
97 72 45
|
syl2anc |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) ) |
99 |
74 72 49
|
syl2anc |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
100 |
99
|
fveq2d |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( (,) ` ( F ` n ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) ) |
101 |
|
df-ov |
|- ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) |
102 |
101
|
a1i |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) ) |
103 |
|
simplr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> n e. ( NN \ A ) ) |
104 |
72 23
|
syldan |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( 1st ` ( F ` n ) ) e. RR* ) |
105 |
104
|
adantr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> ( 1st ` ( F ` n ) ) e. RR* ) |
106 |
72 25
|
syldan |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( 2nd ` ( F ` n ) ) e. RR* ) |
107 |
106
|
adantr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> ( 2nd ` ( F ` n ) ) e. RR* ) |
108 |
|
simpr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) |
109 |
105 107
|
xrltnled |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> ( ( 1st ` ( F ` n ) ) < ( 2nd ` ( F ` n ) ) <-> -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) ) |
110 |
108 109
|
mpbird |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> ( 1st ` ( F ` n ) ) < ( 2nd ` ( F ` n ) ) ) |
111 |
105 107 110
|
xrltled |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) |
112 |
103 111 78
|
syl2anc |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> n e. A ) |
113 |
80
|
ad2antlr |
|- ( ( ( ph /\ n e. ( NN \ A ) ) /\ -. ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) -> -. n e. A ) |
114 |
112 113
|
condan |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) |
115 |
|
ioo0 |
|- ( ( ( 1st ` ( F ` n ) ) e. RR* /\ ( 2nd ` ( F ` n ) ) e. RR* ) -> ( ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = (/) <-> ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) ) |
116 |
104 106 115
|
syl2anc |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = (/) <-> ( 2nd ` ( F ` n ) ) <_ ( 1st ` ( F ` n ) ) ) ) |
117 |
114 116
|
mpbird |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = (/) ) |
118 |
102 117
|
eqtr3d |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) = (/) ) |
119 |
98 100 118
|
3eqtrd |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. F ) ` n ) = (/) ) |
120 |
119
|
iuneq2dv |
|- ( ph -> U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) = U_ n e. ( NN \ A ) (/) ) |
121 |
120 95
|
eqtrd |
|- ( ph -> U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) = (/) ) |
122 |
96 121
|
eqtr4d |
|- ( ph -> U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) = U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) |
123 |
69 122
|
uneq12d |
|- ( ph -> ( U_ n e. A ( ( (,) o. G ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. G ) ` n ) ) = ( U_ n e. A ( ( (,) o. F ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) ) |
124 |
34 38 123
|
3eqtrrd |
|- ( ph -> ( U_ n e. A ( ( (,) o. F ) ` n ) u. U_ n e. ( NN \ A ) ( ( (,) o. F ) ` n ) ) = U. ran ( (,) o. G ) ) |
125 |
11 20 124
|
3eqtrd |
|- ( ph -> U. ran ( (,) o. F ) = U. ran ( (,) o. G ) ) |
126 |
|
volf |
|- vol : dom vol --> ( 0 [,] +oo ) |
127 |
126
|
a1i |
|- ( ph -> vol : dom vol --> ( 0 [,] +oo ) ) |
128 |
1
|
adantr |
|- ( ( ph /\ n e. NN ) -> F : NN --> ( RR* X. RR* ) ) |
129 |
|
simpr |
|- ( ( ph /\ n e. NN ) -> n e. NN ) |
130 |
128 129 45
|
syl2anc |
|- ( ( ph /\ n e. NN ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) ) |
131 |
49
|
fveq2d |
|- ( ( ph /\ n e. NN ) -> ( (,) ` ( F ` n ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) ) |
132 |
101
|
eqcomi |
|- ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) |
133 |
132
|
a1i |
|- ( ( ph /\ n e. NN ) -> ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) ) |
134 |
130 131 133
|
3eqtrd |
|- ( ( ph /\ n e. NN ) -> ( ( (,) o. F ) ` n ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) ) |
135 |
|
ioombl |
|- ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) e. dom vol |
136 |
135
|
a1i |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) e. dom vol ) |
137 |
134 136
|
eqeltrd |
|- ( ( ph /\ n e. NN ) -> ( ( (,) o. F ) ` n ) e. dom vol ) |
138 |
137
|
ralrimiva |
|- ( ph -> A. n e. NN ( ( (,) o. F ) ` n ) e. dom vol ) |
139 |
8 138
|
jca |
|- ( ph -> ( ( (,) o. F ) Fn NN /\ A. n e. NN ( ( (,) o. F ) ` n ) e. dom vol ) ) |
140 |
|
ffnfv |
|- ( ( (,) o. F ) : NN --> dom vol <-> ( ( (,) o. F ) Fn NN /\ A. n e. NN ( ( (,) o. F ) ` n ) e. dom vol ) ) |
141 |
139 140
|
sylibr |
|- ( ph -> ( (,) o. F ) : NN --> dom vol ) |
142 |
|
fco |
|- ( ( vol : dom vol --> ( 0 [,] +oo ) /\ ( (,) o. F ) : NN --> dom vol ) -> ( vol o. ( (,) o. F ) ) : NN --> ( 0 [,] +oo ) ) |
143 |
127 141 142
|
syl2anc |
|- ( ph -> ( vol o. ( (,) o. F ) ) : NN --> ( 0 [,] +oo ) ) |
144 |
143
|
ffnd |
|- ( ph -> ( vol o. ( (,) o. F ) ) Fn NN ) |
145 |
68
|
adantlr |
|- ( ( ( ph /\ n e. NN ) /\ n e. A ) -> ( ( (,) o. G ) ` n ) = ( ( (,) o. F ) ` n ) ) |
146 |
137
|
adantr |
|- ( ( ( ph /\ n e. NN ) /\ n e. A ) -> ( ( (,) o. F ) ` n ) e. dom vol ) |
147 |
145 146
|
eqeltrd |
|- ( ( ( ph /\ n e. NN ) /\ n e. A ) -> ( ( (,) o. G ) ` n ) e. dom vol ) |
148 |
|
simpll |
|- ( ( ( ph /\ n e. NN ) /\ -. n e. A ) -> ph ) |
149 |
|
eldif |
|- ( n e. ( NN \ A ) <-> ( n e. NN /\ -. n e. A ) ) |
150 |
149
|
bicomi |
|- ( ( n e. NN /\ -. n e. A ) <-> n e. ( NN \ A ) ) |
151 |
150
|
biimpi |
|- ( ( n e. NN /\ -. n e. A ) -> n e. ( NN \ A ) ) |
152 |
151
|
adantll |
|- ( ( ( ph /\ n e. NN ) /\ -. n e. A ) -> n e. ( NN \ A ) ) |
153 |
117 135
|
eqeltrrdi |
|- ( ( ph /\ n e. ( NN \ A ) ) -> (/) e. dom vol ) |
154 |
92 153
|
eqeltrd |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. G ) ` n ) e. dom vol ) |
155 |
148 152 154
|
syl2anc |
|- ( ( ( ph /\ n e. NN ) /\ -. n e. A ) -> ( ( (,) o. G ) ` n ) e. dom vol ) |
156 |
147 155
|
pm2.61dan |
|- ( ( ph /\ n e. NN ) -> ( ( (,) o. G ) ` n ) e. dom vol ) |
157 |
156
|
ralrimiva |
|- ( ph -> A. n e. NN ( ( (,) o. G ) ` n ) e. dom vol ) |
158 |
31 157
|
jca |
|- ( ph -> ( ( (,) o. G ) Fn NN /\ A. n e. NN ( ( (,) o. G ) ` n ) e. dom vol ) ) |
159 |
|
ffnfv |
|- ( ( (,) o. G ) : NN --> dom vol <-> ( ( (,) o. G ) Fn NN /\ A. n e. NN ( ( (,) o. G ) ` n ) e. dom vol ) ) |
160 |
158 159
|
sylibr |
|- ( ph -> ( (,) o. G ) : NN --> dom vol ) |
161 |
|
fco |
|- ( ( vol : dom vol --> ( 0 [,] +oo ) /\ ( (,) o. G ) : NN --> dom vol ) -> ( vol o. ( (,) o. G ) ) : NN --> ( 0 [,] +oo ) ) |
162 |
127 160 161
|
syl2anc |
|- ( ph -> ( vol o. ( (,) o. G ) ) : NN --> ( 0 [,] +oo ) ) |
163 |
162
|
ffnd |
|- ( ph -> ( vol o. ( (,) o. G ) ) Fn NN ) |
164 |
145
|
eqcomd |
|- ( ( ( ph /\ n e. NN ) /\ n e. A ) -> ( ( (,) o. F ) ` n ) = ( ( (,) o. G ) ` n ) ) |
165 |
119 92
|
eqtr4d |
|- ( ( ph /\ n e. ( NN \ A ) ) -> ( ( (,) o. F ) ` n ) = ( ( (,) o. G ) ` n ) ) |
166 |
148 152 165
|
syl2anc |
|- ( ( ( ph /\ n e. NN ) /\ -. n e. A ) -> ( ( (,) o. F ) ` n ) = ( ( (,) o. G ) ` n ) ) |
167 |
164 166
|
pm2.61dan |
|- ( ( ph /\ n e. NN ) -> ( ( (,) o. F ) ` n ) = ( ( (,) o. G ) ` n ) ) |
168 |
167
|
fveq2d |
|- ( ( ph /\ n e. NN ) -> ( vol ` ( ( (,) o. F ) ` n ) ) = ( vol ` ( ( (,) o. G ) ` n ) ) ) |
169 |
|
fnfun |
|- ( ( (,) o. F ) Fn NN -> Fun ( (,) o. F ) ) |
170 |
8 169
|
syl |
|- ( ph -> Fun ( (,) o. F ) ) |
171 |
170
|
adantr |
|- ( ( ph /\ n e. NN ) -> Fun ( (,) o. F ) ) |
172 |
7
|
fdmd |
|- ( ph -> dom ( (,) o. F ) = NN ) |
173 |
172
|
eqcomd |
|- ( ph -> NN = dom ( (,) o. F ) ) |
174 |
173
|
adantr |
|- ( ( ph /\ n e. NN ) -> NN = dom ( (,) o. F ) ) |
175 |
129 174
|
eleqtrd |
|- ( ( ph /\ n e. NN ) -> n e. dom ( (,) o. F ) ) |
176 |
|
fvco |
|- ( ( Fun ( (,) o. F ) /\ n e. dom ( (,) o. F ) ) -> ( ( vol o. ( (,) o. F ) ) ` n ) = ( vol ` ( ( (,) o. F ) ` n ) ) ) |
177 |
171 175 176
|
syl2anc |
|- ( ( ph /\ n e. NN ) -> ( ( vol o. ( (,) o. F ) ) ` n ) = ( vol ` ( ( (,) o. F ) ` n ) ) ) |
178 |
|
fnfun |
|- ( ( (,) o. G ) Fn NN -> Fun ( (,) o. G ) ) |
179 |
31 178
|
syl |
|- ( ph -> Fun ( (,) o. G ) ) |
180 |
179
|
adantr |
|- ( ( ph /\ n e. NN ) -> Fun ( (,) o. G ) ) |
181 |
30
|
fdmd |
|- ( ph -> dom ( (,) o. G ) = NN ) |
182 |
181
|
eqcomd |
|- ( ph -> NN = dom ( (,) o. G ) ) |
183 |
182
|
adantr |
|- ( ( ph /\ n e. NN ) -> NN = dom ( (,) o. G ) ) |
184 |
129 183
|
eleqtrd |
|- ( ( ph /\ n e. NN ) -> n e. dom ( (,) o. G ) ) |
185 |
|
fvco |
|- ( ( Fun ( (,) o. G ) /\ n e. dom ( (,) o. G ) ) -> ( ( vol o. ( (,) o. G ) ) ` n ) = ( vol ` ( ( (,) o. G ) ` n ) ) ) |
186 |
180 184 185
|
syl2anc |
|- ( ( ph /\ n e. NN ) -> ( ( vol o. ( (,) o. G ) ) ` n ) = ( vol ` ( ( (,) o. G ) ` n ) ) ) |
187 |
168 177 186
|
3eqtr4d |
|- ( ( ph /\ n e. NN ) -> ( ( vol o. ( (,) o. F ) ) ` n ) = ( ( vol o. ( (,) o. G ) ) ` n ) ) |
188 |
144 163 187
|
eqfnfvd |
|- ( ph -> ( vol o. ( (,) o. F ) ) = ( vol o. ( (,) o. G ) ) ) |
189 |
125 188
|
jca |
|- ( ph -> ( U. ran ( (,) o. F ) = U. ran ( (,) o. G ) /\ ( vol o. ( (,) o. F ) ) = ( vol o. ( (,) o. G ) ) ) ) |