Step |
Hyp |
Ref |
Expression |
1 |
|
ressprdsds.y |
|- ( ph -> Y = ( S Xs_ ( x e. I |-> R ) ) ) |
2 |
|
ressprdsds.h |
|- ( ph -> H = ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) |
3 |
|
ressprdsds.b |
|- B = ( Base ` H ) |
4 |
|
ressprdsds.d |
|- D = ( dist ` Y ) |
5 |
|
ressprdsds.e |
|- E = ( dist ` H ) |
6 |
|
ressprdsds.s |
|- ( ph -> S e. U ) |
7 |
|
ressprdsds.t |
|- ( ph -> T e. V ) |
8 |
|
ressprdsds.i |
|- ( ph -> I e. W ) |
9 |
|
ressprdsds.r |
|- ( ( ph /\ x e. I ) -> R e. X ) |
10 |
|
ressprdsds.a |
|- ( ( ph /\ x e. I ) -> A e. Z ) |
11 |
|
ovres |
|- ( ( f e. B /\ g e. B ) -> ( f ( D |` ( B X. B ) ) g ) = ( f D g ) ) |
12 |
11
|
adantl |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f ( D |` ( B X. B ) ) g ) = ( f D g ) ) |
13 |
|
eqid |
|- ( R |`s A ) = ( R |`s A ) |
14 |
|
eqid |
|- ( dist ` R ) = ( dist ` R ) |
15 |
13 14
|
ressds |
|- ( A e. Z -> ( dist ` R ) = ( dist ` ( R |`s A ) ) ) |
16 |
10 15
|
syl |
|- ( ( ph /\ x e. I ) -> ( dist ` R ) = ( dist ` ( R |`s A ) ) ) |
17 |
16
|
oveqd |
|- ( ( ph /\ x e. I ) -> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) = ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) |
18 |
17
|
mpteq2dva |
|- ( ph -> ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) = ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) ) |
19 |
18
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) = ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) ) |
20 |
19
|
rneqd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ran ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) = ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) ) |
21 |
20
|
uneq1d |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( ran ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) u. { 0 } ) = ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) u. { 0 } ) ) |
22 |
21
|
supeq1d |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) = sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
23 |
|
eqid |
|- ( S Xs_ ( x e. I |-> R ) ) = ( S Xs_ ( x e. I |-> R ) ) |
24 |
|
eqid |
|- ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) = ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) |
25 |
6
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> S e. U ) |
26 |
8
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> I e. W ) |
27 |
9
|
ralrimiva |
|- ( ph -> A. x e. I R e. X ) |
28 |
27
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> A. x e. I R e. X ) |
29 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
30 |
13 29
|
ressbasss |
|- ( Base ` ( R |`s A ) ) C_ ( Base ` R ) |
31 |
30
|
a1i |
|- ( ( ph /\ x e. I ) -> ( Base ` ( R |`s A ) ) C_ ( Base ` R ) ) |
32 |
31
|
ralrimiva |
|- ( ph -> A. x e. I ( Base ` ( R |`s A ) ) C_ ( Base ` R ) ) |
33 |
|
ss2ixp |
|- ( A. x e. I ( Base ` ( R |`s A ) ) C_ ( Base ` R ) -> X_ x e. I ( Base ` ( R |`s A ) ) C_ X_ x e. I ( Base ` R ) ) |
34 |
32 33
|
syl |
|- ( ph -> X_ x e. I ( Base ` ( R |`s A ) ) C_ X_ x e. I ( Base ` R ) ) |
35 |
|
eqid |
|- ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) = ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) |
36 |
|
eqid |
|- ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) = ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) |
37 |
|
ovex |
|- ( R |`s A ) e. _V |
38 |
37
|
rgenw |
|- A. x e. I ( R |`s A ) e. _V |
39 |
38
|
a1i |
|- ( ph -> A. x e. I ( R |`s A ) e. _V ) |
40 |
|
eqid |
|- ( Base ` ( R |`s A ) ) = ( Base ` ( R |`s A ) ) |
41 |
35 36 7 8 39 40
|
prdsbas3 |
|- ( ph -> ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) = X_ x e. I ( Base ` ( R |`s A ) ) ) |
42 |
23 24 6 8 27 29
|
prdsbas3 |
|- ( ph -> ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) = X_ x e. I ( Base ` R ) ) |
43 |
34 41 42
|
3sstr4d |
|- ( ph -> ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) C_ ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
44 |
2
|
fveq2d |
|- ( ph -> ( Base ` H ) = ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
45 |
3 44
|
eqtrid |
|- ( ph -> B = ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
46 |
1
|
fveq2d |
|- ( ph -> ( Base ` Y ) = ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
47 |
43 45 46
|
3sstr4d |
|- ( ph -> B C_ ( Base ` Y ) ) |
48 |
47
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> B C_ ( Base ` Y ) ) |
49 |
46
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( Base ` Y ) = ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
50 |
48 49
|
sseqtrd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> B C_ ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
51 |
|
simprl |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> f e. B ) |
52 |
50 51
|
sseldd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> f e. ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
53 |
|
simprr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> g e. B ) |
54 |
50 53
|
sseldd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> g e. ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
55 |
|
eqid |
|- ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) = ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) |
56 |
23 24 25 26 28 52 54 14 55
|
prdsdsval2 |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) g ) = sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` R ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
57 |
7
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> T e. V ) |
58 |
38
|
a1i |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> A. x e. I ( R |`s A ) e. _V ) |
59 |
45
|
adantr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> B = ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
60 |
51 59
|
eleqtrd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> f e. ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
61 |
53 59
|
eleqtrd |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> g e. ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
62 |
|
eqid |
|- ( dist ` ( R |`s A ) ) = ( dist ` ( R |`s A ) ) |
63 |
|
eqid |
|- ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) = ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) |
64 |
35 36 57 26 58 60 61 62 63
|
prdsdsval2 |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) g ) = sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R |`s A ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
65 |
22 56 64
|
3eqtr4d |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) g ) = ( f ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) g ) ) |
66 |
1
|
fveq2d |
|- ( ph -> ( dist ` Y ) = ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
67 |
4 66
|
eqtrid |
|- ( ph -> D = ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) ) |
68 |
67
|
oveqdr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f D g ) = ( f ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) g ) ) |
69 |
2
|
fveq2d |
|- ( ph -> ( dist ` H ) = ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
70 |
5 69
|
eqtrid |
|- ( ph -> E = ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) |
71 |
70
|
oveqdr |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f E g ) = ( f ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) g ) ) |
72 |
65 68 71
|
3eqtr4d |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f D g ) = ( f E g ) ) |
73 |
12 72
|
eqtr2d |
|- ( ( ph /\ ( f e. B /\ g e. B ) ) -> ( f E g ) = ( f ( D |` ( B X. B ) ) g ) ) |
74 |
73
|
ralrimivva |
|- ( ph -> A. f e. B A. g e. B ( f E g ) = ( f ( D |` ( B X. B ) ) g ) ) |
75 |
8
|
mptexd |
|- ( ph -> ( x e. I |-> ( R |`s A ) ) e. _V ) |
76 |
|
eqid |
|- ( x e. I |-> ( R |`s A ) ) = ( x e. I |-> ( R |`s A ) ) |
77 |
37 76
|
dmmpti |
|- dom ( x e. I |-> ( R |`s A ) ) = I |
78 |
77
|
a1i |
|- ( ph -> dom ( x e. I |-> ( R |`s A ) ) = I ) |
79 |
35 7 75 36 78 63
|
prdsdsfn |
|- ( ph -> ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) Fn ( ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) X. ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) ) |
80 |
45
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) X. ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) ) |
81 |
70 80
|
fneq12d |
|- ( ph -> ( E Fn ( B X. B ) <-> ( dist ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) Fn ( ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) X. ( Base ` ( T Xs_ ( x e. I |-> ( R |`s A ) ) ) ) ) ) ) |
82 |
79 81
|
mpbird |
|- ( ph -> E Fn ( B X. B ) ) |
83 |
8
|
mptexd |
|- ( ph -> ( x e. I |-> R ) e. _V ) |
84 |
|
dmmptg |
|- ( A. x e. I R e. X -> dom ( x e. I |-> R ) = I ) |
85 |
27 84
|
syl |
|- ( ph -> dom ( x e. I |-> R ) = I ) |
86 |
23 6 83 24 85 55
|
prdsdsfn |
|- ( ph -> ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) Fn ( ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) X. ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) ) |
87 |
46
|
sqxpeqd |
|- ( ph -> ( ( Base ` Y ) X. ( Base ` Y ) ) = ( ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) X. ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) ) |
88 |
67 87
|
fneq12d |
|- ( ph -> ( D Fn ( ( Base ` Y ) X. ( Base ` Y ) ) <-> ( dist ` ( S Xs_ ( x e. I |-> R ) ) ) Fn ( ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) X. ( Base ` ( S Xs_ ( x e. I |-> R ) ) ) ) ) ) |
89 |
86 88
|
mpbird |
|- ( ph -> D Fn ( ( Base ` Y ) X. ( Base ` Y ) ) ) |
90 |
|
xpss12 |
|- ( ( B C_ ( Base ` Y ) /\ B C_ ( Base ` Y ) ) -> ( B X. B ) C_ ( ( Base ` Y ) X. ( Base ` Y ) ) ) |
91 |
47 47 90
|
syl2anc |
|- ( ph -> ( B X. B ) C_ ( ( Base ` Y ) X. ( Base ` Y ) ) ) |
92 |
|
fnssres |
|- ( ( D Fn ( ( Base ` Y ) X. ( Base ` Y ) ) /\ ( B X. B ) C_ ( ( Base ` Y ) X. ( Base ` Y ) ) ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) ) |
93 |
89 91 92
|
syl2anc |
|- ( ph -> ( D |` ( B X. B ) ) Fn ( B X. B ) ) |
94 |
|
eqfnov2 |
|- ( ( E Fn ( B X. B ) /\ ( D |` ( B X. B ) ) Fn ( B X. B ) ) -> ( E = ( D |` ( B X. B ) ) <-> A. f e. B A. g e. B ( f E g ) = ( f ( D |` ( B X. B ) ) g ) ) ) |
95 |
82 93 94
|
syl2anc |
|- ( ph -> ( E = ( D |` ( B X. B ) ) <-> A. f e. B A. g e. B ( f E g ) = ( f ( D |` ( B X. B ) ) g ) ) ) |
96 |
74 95
|
mpbird |
|- ( ph -> E = ( D |` ( B X. B ) ) ) |