Step |
Hyp |
Ref |
Expression |
1 |
|
rngchomrnghmresALTV.c |
⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 ) |
2 |
|
rngchomrnghmresALTV.b |
⊢ 𝐵 = ( Rng ∩ 𝑈 ) |
3 |
|
rngchomrnghmresALTV.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
4 |
|
rngchomrnghmresALTV.f |
⊢ 𝐹 = ( Homf ‘ 𝐶 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
6 |
1 5 3
|
rngcbasALTV |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
7 |
|
inss2 |
⊢ ( 𝑈 ∩ Rng ) ⊆ Rng |
8 |
6 7
|
eqsstrdi |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) ⊆ Rng ) |
9 |
|
resmpo |
⊢ ( ( ( Base ‘ 𝐶 ) ⊆ Rng ∧ ( Base ‘ 𝐶 ) ⊆ Rng ) → ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) ) |
10 |
8 8 9
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) ) |
11 |
|
df-rnghomo |
⊢ RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ) |
12 |
|
ovex |
⊢ ( 𝑤 ↑m 𝑣 ) ∈ V |
13 |
12
|
rabex |
⊢ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
14 |
13
|
csbex |
⊢ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
15 |
14
|
csbex |
⊢ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
16 |
11 15
|
fnmpoi |
⊢ RngHomo Fn ( Rng × Rng ) |
17 |
16
|
a1i |
⊢ ( 𝜑 → RngHomo Fn ( Rng × Rng ) ) |
18 |
|
fnov |
⊢ ( RngHomo Fn ( Rng × Rng ) ↔ RngHomo = ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ) |
19 |
17 18
|
sylib |
⊢ ( 𝜑 → RngHomo = ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ) |
20 |
|
incom |
⊢ ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) |
21 |
20
|
a1i |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) ) |
22 |
2
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Rng ∩ 𝑈 ) ) |
23 |
21 6 22
|
3eqtr4rd |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) ) |
24 |
23
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) |
25 |
19 24
|
reseq12d |
⊢ ( 𝜑 → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) ) |
26 |
1 5 3 4
|
rngchomffvalALTV |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) ) |
27 |
10 25 26
|
3eqtr4rd |
⊢ ( 𝜑 → 𝐹 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) ) |