Step |
Hyp |
Ref |
Expression |
1 |
|
hofcl.m |
|- M = ( HomF ` C ) |
2 |
|
hofcl.o |
|- O = ( oppCat ` C ) |
3 |
|
hofcl.d |
|- D = ( SetCat ` U ) |
4 |
|
hofcl.c |
|- ( ph -> C e. Cat ) |
5 |
|
hofcl.u |
|- ( ph -> U e. V ) |
6 |
|
hofcl.h |
|- ( ph -> ran ( Homf ` C ) C_ U ) |
7 |
|
hofcllem.b |
|- B = ( Base ` C ) |
8 |
|
hofcllem.h |
|- H = ( Hom ` C ) |
9 |
|
hofcllem.x |
|- ( ph -> X e. B ) |
10 |
|
hofcllem.y |
|- ( ph -> Y e. B ) |
11 |
|
hofcllem.z |
|- ( ph -> Z e. B ) |
12 |
|
hofcllem.w |
|- ( ph -> W e. B ) |
13 |
|
hofcllem.s |
|- ( ph -> S e. B ) |
14 |
|
hofcllem.t |
|- ( ph -> T e. B ) |
15 |
|
hofcllem.m |
|- ( ph -> K e. ( Z H X ) ) |
16 |
|
hofcllem.n |
|- ( ph -> L e. ( Y H W ) ) |
17 |
|
hofcllem.p |
|- ( ph -> P e. ( S H Z ) ) |
18 |
|
hofcllem.q |
|- ( ph -> Q e. ( W H T ) ) |
19 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
20 |
4
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> C e. Cat ) |
21 |
13
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> S e. B ) |
22 |
11
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> Z e. B ) |
23 |
9
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> X e. B ) |
24 |
17
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> P e. ( S H Z ) ) |
25 |
15
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> K e. ( Z H X ) ) |
26 |
14
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> T e. B ) |
27 |
10
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> Y e. B ) |
28 |
|
simpr |
|- ( ( ph /\ f e. ( X H Y ) ) -> f e. ( X H Y ) ) |
29 |
7 8 19 4 10 12 14 16 18
|
catcocl |
|- ( ph -> ( Q ( <. Y , W >. ( comp ` C ) T ) L ) e. ( Y H T ) ) |
30 |
29
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( Q ( <. Y , W >. ( comp ` C ) T ) L ) e. ( Y H T ) ) |
31 |
7 8 19 20 23 27 26 28 30
|
catcocl |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) e. ( X H T ) ) |
32 |
7 8 19 20 21 22 23 24 25 26 31
|
catass |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. Z , X >. ( comp ` C ) T ) K ) ( <. S , Z >. ( comp ` C ) T ) P ) = ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. S , X >. ( comp ` C ) T ) ( K ( <. S , Z >. ( comp ` C ) X ) P ) ) ) |
33 |
12
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> W e. B ) |
34 |
16
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> L e. ( Y H W ) ) |
35 |
18
|
adantr |
|- ( ( ph /\ f e. ( X H Y ) ) -> Q e. ( W H T ) ) |
36 |
7 8 19 20 23 27 33 28 34 26 35
|
catass |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) = ( Q ( <. X , W >. ( comp ` C ) T ) ( L ( <. X , Y >. ( comp ` C ) W ) f ) ) ) |
37 |
36
|
oveq1d |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. Z , X >. ( comp ` C ) T ) K ) = ( ( Q ( <. X , W >. ( comp ` C ) T ) ( L ( <. X , Y >. ( comp ` C ) W ) f ) ) ( <. Z , X >. ( comp ` C ) T ) K ) ) |
38 |
7 8 19 20 23 27 33 28 34
|
catcocl |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( L ( <. X , Y >. ( comp ` C ) W ) f ) e. ( X H W ) ) |
39 |
7 8 19 20 22 23 33 25 38 26 35
|
catass |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( Q ( <. X , W >. ( comp ` C ) T ) ( L ( <. X , Y >. ( comp ` C ) W ) f ) ) ( <. Z , X >. ( comp ` C ) T ) K ) = ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) |
40 |
37 39
|
eqtrd |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. Z , X >. ( comp ` C ) T ) K ) = ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) |
41 |
40
|
oveq1d |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. Z , X >. ( comp ` C ) T ) K ) ( <. S , Z >. ( comp ` C ) T ) P ) = ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) |
42 |
32 41
|
eqtr3d |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. S , X >. ( comp ` C ) T ) ( K ( <. S , Z >. ( comp ` C ) X ) P ) ) = ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) |
43 |
42
|
mpteq2dva |
|- ( ph -> ( f e. ( X H Y ) |-> ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. S , X >. ( comp ` C ) T ) ( K ( <. S , Z >. ( comp ` C ) X ) P ) ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ) |
44 |
7 8 19 4 13 11 9 17 15
|
catcocl |
|- ( ph -> ( K ( <. S , Z >. ( comp ` C ) X ) P ) e. ( S H X ) ) |
45 |
1 4 7 8 9 10 13 14 19 44 29
|
hof2val |
|- ( ph -> ( ( K ( <. S , Z >. ( comp ` C ) X ) P ) ( <. X , Y >. ( 2nd ` M ) <. S , T >. ) ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ) = ( f e. ( X H Y ) |-> ( ( ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ( <. X , Y >. ( comp ` C ) T ) f ) ( <. S , X >. ( comp ` C ) T ) ( K ( <. S , Z >. ( comp ` C ) X ) P ) ) ) ) |
46 |
1 4 7 8 11 12 13 14 19 17 18
|
hof2val |
|- ( ph -> ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) = ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ) |
47 |
1 4 7 8 9 10 11 12 19 15 16
|
hof2val |
|- ( ph -> ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) = ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) |
48 |
46 47
|
oveq12d |
|- ( ph -> ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. ( comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) = ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ( <. ( X H Y ) , ( Z H W ) >. ( comp ` D ) ( S H T ) ) ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) ) |
49 |
|
eqid |
|- ( comp ` D ) = ( comp ` D ) |
50 |
|
eqid |
|- ( Homf ` C ) = ( Homf ` C ) |
51 |
50 7 8 9 10
|
homfval |
|- ( ph -> ( X ( Homf ` C ) Y ) = ( X H Y ) ) |
52 |
50 7
|
homffn |
|- ( Homf ` C ) Fn ( B X. B ) |
53 |
52
|
a1i |
|- ( ph -> ( Homf ` C ) Fn ( B X. B ) ) |
54 |
|
df-f |
|- ( ( Homf ` C ) : ( B X. B ) --> U <-> ( ( Homf ` C ) Fn ( B X. B ) /\ ran ( Homf ` C ) C_ U ) ) |
55 |
53 6 54
|
sylanbrc |
|- ( ph -> ( Homf ` C ) : ( B X. B ) --> U ) |
56 |
55 9 10
|
fovrnd |
|- ( ph -> ( X ( Homf ` C ) Y ) e. U ) |
57 |
51 56
|
eqeltrrd |
|- ( ph -> ( X H Y ) e. U ) |
58 |
50 7 8 11 12
|
homfval |
|- ( ph -> ( Z ( Homf ` C ) W ) = ( Z H W ) ) |
59 |
55 11 12
|
fovrnd |
|- ( ph -> ( Z ( Homf ` C ) W ) e. U ) |
60 |
58 59
|
eqeltrrd |
|- ( ph -> ( Z H W ) e. U ) |
61 |
50 7 8 13 14
|
homfval |
|- ( ph -> ( S ( Homf ` C ) T ) = ( S H T ) ) |
62 |
55 13 14
|
fovrnd |
|- ( ph -> ( S ( Homf ` C ) T ) e. U ) |
63 |
61 62
|
eqeltrrd |
|- ( ph -> ( S H T ) e. U ) |
64 |
7 8 19 20 22 23 33 25 38
|
catcocl |
|- ( ( ph /\ f e. ( X H Y ) ) -> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) e. ( Z H W ) ) |
65 |
64
|
fmpttd |
|- ( ph -> ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) : ( X H Y ) --> ( Z H W ) ) |
66 |
4
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> C e. Cat ) |
67 |
13
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> S e. B ) |
68 |
11
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> Z e. B ) |
69 |
14
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> T e. B ) |
70 |
17
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> P e. ( S H Z ) ) |
71 |
12
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> W e. B ) |
72 |
|
simpr |
|- ( ( ph /\ g e. ( Z H W ) ) -> g e. ( Z H W ) ) |
73 |
18
|
adantr |
|- ( ( ph /\ g e. ( Z H W ) ) -> Q e. ( W H T ) ) |
74 |
7 8 19 66 68 71 69 72 73
|
catcocl |
|- ( ( ph /\ g e. ( Z H W ) ) -> ( Q ( <. Z , W >. ( comp ` C ) T ) g ) e. ( Z H T ) ) |
75 |
7 8 19 66 67 68 69 70 74
|
catcocl |
|- ( ( ph /\ g e. ( Z H W ) ) -> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) e. ( S H T ) ) |
76 |
75
|
fmpttd |
|- ( ph -> ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) : ( Z H W ) --> ( S H T ) ) |
77 |
3 5 49 57 60 63 65 76
|
setcco |
|- ( ph -> ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ( <. ( X H Y ) , ( Z H W ) >. ( comp ` D ) ( S H T ) ) ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) = ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) o. ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) ) |
78 |
|
eqidd |
|- ( ph -> ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) = ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) |
79 |
|
eqidd |
|- ( ph -> ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) = ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ) |
80 |
|
oveq2 |
|- ( g = ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) -> ( Q ( <. Z , W >. ( comp ` C ) T ) g ) = ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) |
81 |
80
|
oveq1d |
|- ( g = ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) -> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) = ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) |
82 |
64 78 79 81
|
fmptco |
|- ( ph -> ( ( g e. ( Z H W ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) g ) ( <. S , Z >. ( comp ` C ) T ) P ) ) o. ( f e. ( X H Y ) |-> ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ) |
83 |
48 77 82
|
3eqtrd |
|- ( ph -> ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. ( comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) = ( f e. ( X H Y ) |-> ( ( Q ( <. Z , W >. ( comp ` C ) T ) ( ( L ( <. X , Y >. ( comp ` C ) W ) f ) ( <. Z , X >. ( comp ` C ) W ) K ) ) ( <. S , Z >. ( comp ` C ) T ) P ) ) ) |
84 |
43 45 83
|
3eqtr4d |
|- ( ph -> ( ( K ( <. S , Z >. ( comp ` C ) X ) P ) ( <. X , Y >. ( 2nd ` M ) <. S , T >. ) ( Q ( <. Y , W >. ( comp ` C ) T ) L ) ) = ( ( P ( <. Z , W >. ( 2nd ` M ) <. S , T >. ) Q ) ( <. ( X H Y ) , ( Z H W ) >. ( comp ` D ) ( S H T ) ) ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) ) |