Step |
Hyp |
Ref |
Expression |
1 |
|
hofval.m |
⊢ 𝑀 = ( HomF ‘ 𝐶 ) |
2 |
|
hofval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
3 |
|
hof1.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
4 |
|
hof1.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
5 |
|
hof1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
hof1.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
hof2.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
8 |
|
hof2.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝐵 ) |
9 |
|
hof2.o |
⊢ · = ( comp ‘ 𝐶 ) |
10 |
1 2 3 4 9
|
hofval |
⊢ ( 𝜑 → 𝑀 = 〈 ( Homf ‘ 𝐶 ) , ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ ( 𝐵 × 𝐵 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) 〉 ) |
11 |
|
fvex |
⊢ ( Homf ‘ 𝐶 ) ∈ V |
12 |
3
|
fvexi |
⊢ 𝐵 ∈ V |
13 |
12 12
|
xpex |
⊢ ( 𝐵 × 𝐵 ) ∈ V |
14 |
13 13
|
mpoex |
⊢ ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ ( 𝐵 × 𝐵 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) ∈ V |
15 |
11 14
|
op2ndd |
⊢ ( 𝑀 = 〈 ( Homf ‘ 𝐶 ) , ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ ( 𝐵 × 𝐵 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) 〉 → ( 2nd ‘ 𝑀 ) = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ ( 𝐵 × 𝐵 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) ) |
16 |
10 15
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 𝑀 ) = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ ( 𝐵 × 𝐵 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) ) |
17 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → 𝑦 = 〈 𝑍 , 𝑊 〉 ) |
18 |
17
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 𝑦 ) = ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) ) |
19 |
|
op1stg |
⊢ ( ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
20 |
7 8 19
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
21 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
22 |
18 21
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 𝑦 ) = 𝑍 ) |
23 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → 𝑥 = 〈 𝑋 , 𝑌 〉 ) |
24 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 𝑥 ) = ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) ) |
25 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
26 |
5 6 25
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
28 |
24 27
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 1st ‘ 𝑥 ) = 𝑋 ) |
29 |
22 28
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) = ( 𝑍 𝐻 𝑋 ) ) |
30 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) ) |
31 |
|
op2ndg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
32 |
5 6 31
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
34 |
30 33
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 𝑥 ) = 𝑌 ) |
35 |
17
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 𝑦 ) = ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) ) |
36 |
|
op2ndg |
⊢ ( ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
37 |
7 8 36
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
39 |
35 38
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 2nd ‘ 𝑦 ) = 𝑊 ) |
40 |
34 39
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) = ( 𝑌 𝐻 𝑊 ) ) |
41 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 〈 𝑋 , 𝑌 〉 ) ) |
42 |
|
df-ov |
⊢ ( 𝑋 𝐻 𝑌 ) = ( 𝐻 ‘ 〈 𝑋 , 𝑌 〉 ) |
43 |
41 42
|
eqtr4di |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 𝐻 ‘ 𝑥 ) = ( 𝑋 𝐻 𝑌 ) ) |
44 |
22 28
|
opeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 = 〈 𝑍 , 𝑋 〉 ) |
45 |
44 39
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) = ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) ) |
46 |
23 39
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 𝑥 · ( 2nd ‘ 𝑦 ) ) = ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ) |
47 |
46
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) = ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ) |
48 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → 𝑓 = 𝑓 ) |
49 |
45 47 48
|
oveq123d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) = ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) |
50 |
43 49
|
mpteq12dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) = ( ℎ ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) ) |
51 |
29 40 50
|
mpoeq123dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 〈 𝑋 , 𝑌 〉 ∧ 𝑦 = 〈 𝑍 , 𝑊 〉 ) ) → ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) 𝐻 ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐻 ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( 𝐻 ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 · ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 · ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) = ( 𝑓 ∈ ( 𝑍 𝐻 𝑋 ) , 𝑔 ∈ ( 𝑌 𝐻 𝑊 ) ↦ ( ℎ ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) ) ) |
52 |
5 6
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐵 × 𝐵 ) ) |
53 |
7 8
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑍 , 𝑊 〉 ∈ ( 𝐵 × 𝐵 ) ) |
54 |
|
ovex |
⊢ ( 𝑍 𝐻 𝑋 ) ∈ V |
55 |
|
ovex |
⊢ ( 𝑌 𝐻 𝑊 ) ∈ V |
56 |
54 55
|
mpoex |
⊢ ( 𝑓 ∈ ( 𝑍 𝐻 𝑋 ) , 𝑔 ∈ ( 𝑌 𝐻 𝑊 ) ↦ ( ℎ ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) ) ∈ V |
57 |
56
|
a1i |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑍 𝐻 𝑋 ) , 𝑔 ∈ ( 𝑌 𝐻 𝑊 ) ↦ ( ℎ ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) ) ∈ V ) |
58 |
16 51 52 53 57
|
ovmpod |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ 𝑀 ) 〈 𝑍 , 𝑊 〉 ) = ( 𝑓 ∈ ( 𝑍 𝐻 𝑋 ) , 𝑔 ∈ ( 𝑌 𝐻 𝑊 ) ↦ ( ℎ ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑊 ) ℎ ) ( 〈 𝑍 , 𝑋 〉 · 𝑊 ) 𝑓 ) ) ) ) |