Step |
Hyp |
Ref |
Expression |
1 |
|
comfeq.1 |
|- .x. = ( comp ` C ) |
2 |
|
comfeq.2 |
|- .xb = ( comp ` D ) |
3 |
|
comfeq.h |
|- H = ( Hom ` C ) |
4 |
|
comfeq.3 |
|- ( ph -> B = ( Base ` C ) ) |
5 |
|
comfeq.4 |
|- ( ph -> B = ( Base ` D ) ) |
6 |
|
comfeq.5 |
|- ( ph -> ( Homf ` C ) = ( Homf ` D ) ) |
7 |
4
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` C ) X. ( Base ` C ) ) ) |
8 |
|
eqidd |
|- ( ph -> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) |
9 |
7 4 8
|
mpoeq123dv |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) ) |
10 |
|
eqid |
|- ( comf ` C ) = ( comf ` C ) |
11 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
12 |
10 11 3 1
|
comfffval |
|- ( comf ` C ) = ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) |
13 |
9 12
|
eqtr4di |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( comf ` C ) ) |
14 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
15 |
6
|
3ad2ant1 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( Homf ` C ) = ( Homf ` D ) ) |
16 |
|
xp2nd |
|- ( u e. ( B X. B ) -> ( 2nd ` u ) e. B ) |
17 |
16
|
3ad2ant2 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 2nd ` u ) e. B ) |
18 |
4
|
3ad2ant1 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> B = ( Base ` C ) ) |
19 |
17 18
|
eleqtrd |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 2nd ` u ) e. ( Base ` C ) ) |
20 |
|
simp3 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> z e. B ) |
21 |
20 18
|
eleqtrd |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> z e. ( Base ` C ) ) |
22 |
11 3 14 15 19 21
|
homfeqval |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( 2nd ` u ) H z ) = ( ( 2nd ` u ) ( Hom ` D ) z ) ) |
23 |
|
xp1st |
|- ( u e. ( B X. B ) -> ( 1st ` u ) e. B ) |
24 |
23
|
3ad2ant2 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 1st ` u ) e. B ) |
25 |
24 18
|
eleqtrd |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 1st ` u ) e. ( Base ` C ) ) |
26 |
11 3 14 15 25 19
|
homfeqval |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( 1st ` u ) H ( 2nd ` u ) ) = ( ( 1st ` u ) ( Hom ` D ) ( 2nd ` u ) ) ) |
27 |
|
df-ov |
|- ( ( 1st ` u ) H ( 2nd ` u ) ) = ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) |
28 |
|
df-ov |
|- ( ( 1st ` u ) ( Hom ` D ) ( 2nd ` u ) ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) |
29 |
26 27 28
|
3eqtr3g |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) ) |
30 |
|
1st2nd2 |
|- ( u e. ( B X. B ) -> u = <. ( 1st ` u ) , ( 2nd ` u ) >. ) |
31 |
30
|
3ad2ant2 |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> u = <. ( 1st ` u ) , ( 2nd ` u ) >. ) |
32 |
31
|
fveq2d |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` u ) = ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) ) |
33 |
31
|
fveq2d |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( Hom ` D ) ` u ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) ) |
34 |
29 32 33
|
3eqtr4d |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` u ) = ( ( Hom ` D ) ` u ) ) |
35 |
|
eqidd |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( g ( u .xb z ) f ) = ( g ( u .xb z ) f ) ) |
36 |
22 34 35
|
mpoeq123dv |
|- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) = ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) |
37 |
36
|
mpoeq3dva |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) ) |
38 |
5
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` D ) X. ( Base ` D ) ) ) |
39 |
|
eqidd |
|- ( ph -> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) = ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) |
40 |
38 5 39
|
mpoeq123dv |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) ) |
41 |
37 40
|
eqtrd |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) ) |
42 |
|
eqid |
|- ( comf ` D ) = ( comf ` D ) |
43 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
44 |
42 43 14 2
|
comfffval |
|- ( comf ` D ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) |
45 |
41 44
|
eqtr4di |
|- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( comf ` D ) ) |
46 |
13 45
|
eqeq12d |
|- ( ph -> ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> ( comf ` C ) = ( comf ` D ) ) ) |
47 |
|
ovex |
|- ( ( 2nd ` u ) H z ) e. _V |
48 |
|
fvex |
|- ( H ` u ) e. _V |
49 |
47 48
|
mpoex |
|- ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V |
50 |
49
|
rgen2w |
|- A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V |
51 |
|
mpo2eqb |
|- ( A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V -> ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) ) |
52 |
50 51
|
ax-mp |
|- ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) |
53 |
|
vex |
|- x e. _V |
54 |
|
vex |
|- y e. _V |
55 |
53 54
|
op2ndd |
|- ( u = <. x , y >. -> ( 2nd ` u ) = y ) |
56 |
55
|
oveq1d |
|- ( u = <. x , y >. -> ( ( 2nd ` u ) H z ) = ( y H z ) ) |
57 |
|
fveq2 |
|- ( u = <. x , y >. -> ( H ` u ) = ( H ` <. x , y >. ) ) |
58 |
|
df-ov |
|- ( x H y ) = ( H ` <. x , y >. ) |
59 |
57 58
|
eqtr4di |
|- ( u = <. x , y >. -> ( H ` u ) = ( x H y ) ) |
60 |
|
oveq1 |
|- ( u = <. x , y >. -> ( u .x. z ) = ( <. x , y >. .x. z ) ) |
61 |
60
|
oveqd |
|- ( u = <. x , y >. -> ( g ( u .x. z ) f ) = ( g ( <. x , y >. .x. z ) f ) ) |
62 |
|
oveq1 |
|- ( u = <. x , y >. -> ( u .xb z ) = ( <. x , y >. .xb z ) ) |
63 |
62
|
oveqd |
|- ( u = <. x , y >. -> ( g ( u .xb z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) |
64 |
61 63
|
eqeq12d |
|- ( u = <. x , y >. -> ( ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |
65 |
59 64
|
raleqbidv |
|- ( u = <. x , y >. -> ( A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |
66 |
56 65
|
raleqbidv |
|- ( u = <. x , y >. -> ( A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> A. g e. ( y H z ) A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |
67 |
|
ovex |
|- ( g ( u .x. z ) f ) e. _V |
68 |
67
|
rgen2w |
|- A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) e. _V |
69 |
|
mpo2eqb |
|- ( A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) e. _V -> ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) ) ) |
70 |
68 69
|
ax-mp |
|- ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) ) |
71 |
|
ralcom |
|- ( A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) <-> A. g e. ( y H z ) A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) |
72 |
66 70 71
|
3bitr4g |
|- ( u = <. x , y >. -> ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |
73 |
72
|
ralbidv |
|- ( u = <. x , y >. -> ( A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |
74 |
73
|
ralxp |
|- ( A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) |
75 |
52 74
|
bitri |
|- ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) |
76 |
46 75
|
bitr3di |
|- ( ph -> ( ( comf ` C ) = ( comf ` D ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) ) |