Step |
Hyp |
Ref |
Expression |
1 |
|
rngcbasALTV.c |
⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 ) |
2 |
|
rngcbasALTV.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
rngcbasALTV.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
4 |
|
rngccofvalALTV.o |
⊢ · = ( comp ‘ 𝐶 ) |
5 |
|
rngccoALTV.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
rngccoALTV.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
rngccoALTV.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
8 |
|
rngccoALTV.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) ) |
9 |
|
rngccoALTV.g |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 RngHomo 𝑍 ) ) |
10 |
1 2 3 4
|
rngccofvalALTV |
⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RngHomo ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → 𝑣 = 〈 𝑋 , 𝑌 〉 ) |
12 |
11
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) ) |
13 |
|
op2ndg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
14 |
5 6 13
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
16 |
12 15
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑣 ) = 𝑌 ) |
17 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → 𝑧 = 𝑍 ) |
18 |
16 17
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( ( 2nd ‘ 𝑣 ) RngHomo 𝑧 ) = ( 𝑌 RngHomo 𝑍 ) ) |
19 |
11
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ 𝑣 ) = ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) ) |
20 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
21 |
5 6 20
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
23 |
19 22
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ 𝑣 ) = 𝑋 ) |
24 |
23 16
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( ( 1st ‘ 𝑣 ) RngHomo ( 2nd ‘ 𝑣 ) ) = ( 𝑋 RngHomo 𝑌 ) ) |
25 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 𝑔 ∘ 𝑓 ) = ( 𝑔 ∘ 𝑓 ) ) |
26 |
18 24 25
|
mpoeq123dv |
⊢ ( ( 𝜑 ∧ ( 𝑣 = 〈 𝑋 , 𝑌 〉 ∧ 𝑧 = 𝑍 ) ) → ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RngHomo ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) = ( 𝑔 ∈ ( 𝑌 RngHomo 𝑍 ) , 𝑓 ∈ ( 𝑋 RngHomo 𝑌 ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) |
27 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐵 × 𝐵 ) ) |
28 |
5 6 27
|
syl2anc |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐵 × 𝐵 ) ) |
29 |
|
ovex |
⊢ ( 𝑌 RngHomo 𝑍 ) ∈ V |
30 |
|
ovex |
⊢ ( 𝑋 RngHomo 𝑌 ) ∈ V |
31 |
29 30
|
mpoex |
⊢ ( 𝑔 ∈ ( 𝑌 RngHomo 𝑍 ) , 𝑓 ∈ ( 𝑋 RngHomo 𝑌 ) ↦ ( 𝑔 ∘ 𝑓 ) ) ∈ V |
32 |
31
|
a1i |
⊢ ( 𝜑 → ( 𝑔 ∈ ( 𝑌 RngHomo 𝑍 ) , 𝑓 ∈ ( 𝑋 RngHomo 𝑌 ) ↦ ( 𝑔 ∘ 𝑓 ) ) ∈ V ) |
33 |
10 26 28 7 32
|
ovmpod |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) = ( 𝑔 ∈ ( 𝑌 RngHomo 𝑍 ) , 𝑓 ∈ ( 𝑋 RngHomo 𝑌 ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) |
34 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) ) → 𝑔 = 𝐺 ) |
35 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) ) → 𝑓 = 𝐹 ) |
36 |
34 35
|
coeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) ) → ( 𝑔 ∘ 𝑓 ) = ( 𝐺 ∘ 𝐹 ) ) |
37 |
|
coexg |
⊢ ( ( 𝐺 ∈ ( 𝑌 RngHomo 𝑍 ) ∧ 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) ) → ( 𝐺 ∘ 𝐹 ) ∈ V ) |
38 |
9 8 37
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ∘ 𝐹 ) ∈ V ) |
39 |
33 36 9 8 38
|
ovmpod |
⊢ ( 𝜑 → ( 𝐺 ( 〈 𝑋 , 𝑌 〉 · 𝑍 ) 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |