| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rngccatALTV.c |
⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 ) |
| 2 |
|
rngccatidALTV.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
| 3 |
2
|
a1i |
⊢ ( 𝑈 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐶 ) ) |
| 4 |
|
eqidd |
⊢ ( 𝑈 ∈ 𝑉 → ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) ) |
| 5 |
|
eqidd |
⊢ ( 𝑈 ∈ 𝑉 → ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) ) |
| 6 |
1
|
fvexi |
⊢ 𝐶 ∈ V |
| 7 |
6
|
a1i |
⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ V ) |
| 8 |
|
biid |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ↔ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) |
| 9 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → 𝑈 ∈ 𝑉 ) |
| 10 |
1 2 9
|
rngcbasALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → 𝐵 = ( 𝑈 ∩ Rng ) ) |
| 11 |
|
eleq2 |
⊢ ( 𝐵 = ( 𝑈 ∩ Rng ) → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ( 𝑈 ∩ Rng ) ) ) |
| 12 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝑈 ∩ Rng ) ↔ ( 𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng ) ) |
| 13 |
12
|
simprbi |
⊢ ( 𝑥 ∈ ( 𝑈 ∩ Rng ) → 𝑥 ∈ Rng ) |
| 14 |
11 13
|
biimtrdi |
⊢ ( 𝐵 = ( 𝑈 ∩ Rng ) → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ Rng ) ) |
| 15 |
14
|
com12 |
⊢ ( 𝑥 ∈ 𝐵 → ( 𝐵 = ( 𝑈 ∩ Rng ) → 𝑥 ∈ Rng ) ) |
| 16 |
15
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐵 = ( 𝑈 ∩ Rng ) → 𝑥 ∈ Rng ) ) |
| 17 |
10 16
|
mpd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ Rng ) |
| 18 |
|
eqid |
⊢ ( Base ‘ 𝑥 ) = ( Base ‘ 𝑥 ) |
| 19 |
18
|
idrnghm |
⊢ ( 𝑥 ∈ Rng → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) |
| 20 |
17 19
|
syl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) |
| 21 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
| 22 |
|
simpr |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
| 23 |
1 2 9 21 22 22
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑥 RngHom 𝑥 ) ) |
| 24 |
20 23
|
eleqtrrd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) |
| 25 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑈 ∈ 𝑉 ) |
| 26 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
| 27 |
|
simpl |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → 𝑤 ∈ 𝐵 ) |
| 28 |
27
|
3ad2ant1 |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → 𝑤 ∈ 𝐵 ) |
| 29 |
28
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑤 ∈ 𝐵 ) |
| 30 |
|
simpr |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
| 31 |
30
|
3ad2ant1 |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → 𝑥 ∈ 𝐵 ) |
| 32 |
31
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑥 ∈ 𝐵 ) |
| 33 |
|
simp1 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑈 ∈ 𝑉 ) |
| 34 |
27
|
3ad2ant3 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑤 ∈ 𝐵 ) |
| 35 |
30
|
3ad2ant3 |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
| 36 |
1 2 33 21 34 35
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑤 RngHom 𝑥 ) ) |
| 37 |
36
|
eleq2d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ↔ 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) |
| 38 |
37
|
biimpd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) |
| 39 |
38
|
3exp |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) ) ) |
| 40 |
39
|
com14 |
⊢ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) ) ) |
| 41 |
40
|
3ad2ant1 |
⊢ ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) ) ) |
| 42 |
41
|
com13 |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) ) ) |
| 43 |
42
|
3imp |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) ) |
| 44 |
43
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) |
| 45 |
20
|
expcom |
⊢ ( 𝑥 ∈ 𝐵 → ( 𝑈 ∈ 𝑉 → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) ) |
| 46 |
45
|
adantl |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) ) |
| 47 |
46
|
3ad2ant1 |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) ) |
| 48 |
47
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) |
| 49 |
1 2 25 26 29 32 32 44 48
|
rngccoALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( I ↾ ( Base ‘ 𝑥 ) ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) ) |
| 50 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑈 ∈ 𝑉 ) |
| 51 |
|
simprl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑤 ∈ 𝐵 ) |
| 52 |
|
simprr |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
| 53 |
1 2 50 21 51 52
|
elrngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → 𝑓 : ( Base ‘ 𝑤 ) ⟶ ( Base ‘ 𝑥 ) ) ) |
| 54 |
53
|
ex |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → 𝑓 : ( Base ‘ 𝑤 ) ⟶ ( Base ‘ 𝑥 ) ) ) ) |
| 55 |
54
|
com13 |
⊢ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → 𝑓 : ( Base ‘ 𝑤 ) ⟶ ( Base ‘ 𝑥 ) ) ) ) |
| 56 |
|
fcoi2 |
⊢ ( 𝑓 : ( Base ‘ 𝑤 ) ⟶ ( Base ‘ 𝑥 ) → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) |
| 57 |
55 56
|
syl8 |
⊢ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) ) ) |
| 58 |
57
|
3ad2ant1 |
⊢ ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) ) ) |
| 59 |
58
|
com12 |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) ) ) |
| 60 |
59
|
a1d |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) ) ) ) |
| 61 |
60
|
3imp |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) ) |
| 62 |
61
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( I ↾ ( Base ‘ 𝑥 ) ) ∘ 𝑓 ) = 𝑓 ) |
| 63 |
49 62
|
eqtrd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( I ↾ ( Base ‘ 𝑥 ) ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ) |
| 64 |
|
simp3 |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → 𝑈 ∈ 𝑉 ) |
| 65 |
30
|
adantr |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
| 66 |
65
|
3ad2ant2 |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → 𝑥 ∈ 𝐵 ) |
| 67 |
|
simprl |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
| 68 |
67
|
3ad2ant2 |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → 𝑦 ∈ 𝐵 ) |
| 69 |
46
|
adantr |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) ) |
| 70 |
69
|
a1i |
⊢ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) ) ) |
| 71 |
70
|
3imp |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RngHom 𝑥 ) ) |
| 72 |
|
simpl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → 𝑈 ∈ 𝑉 ) |
| 73 |
65
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → 𝑥 ∈ 𝐵 ) |
| 74 |
67
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → 𝑦 ∈ 𝐵 ) |
| 75 |
1 2 72 21 73 74
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 RngHom 𝑦 ) ) |
| 76 |
75
|
eleq2d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↔ 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) |
| 77 |
76
|
biimpd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) |
| 78 |
77
|
ex |
⊢ ( 𝑈 ∈ 𝑉 → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) |
| 79 |
78
|
com13 |
⊢ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) |
| 80 |
79
|
3imp |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) |
| 81 |
1 2 64 26 66 66 68 71 80
|
rngccoALTV |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = ( 𝑔 ∘ ( I ↾ ( Base ‘ 𝑥 ) ) ) ) |
| 82 |
1 2 72 21 73 74
|
elrngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) |
| 83 |
82
|
ex |
⊢ ( 𝑈 ∈ 𝑉 → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) ) |
| 84 |
83
|
com13 |
⊢ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → 𝑔 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) ) |
| 85 |
84
|
3imp |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → 𝑔 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) |
| 86 |
|
fcoi1 |
⊢ ( 𝑔 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) → ( 𝑔 ∘ ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) |
| 87 |
85 86
|
syl |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → ( 𝑔 ∘ ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) |
| 88 |
81 87
|
eqtrd |
⊢ ( ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑈 ∈ 𝑉 ) → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) |
| 89 |
88
|
3exp |
⊢ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) ) ) |
| 90 |
89
|
3ad2ant2 |
⊢ ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑈 ∈ 𝑉 → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) ) ) |
| 91 |
90
|
expdcom |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) ) ) ) |
| 92 |
91
|
3imp |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) ) |
| 93 |
92
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) ( I ↾ ( Base ‘ 𝑥 ) ) ) = 𝑔 ) |
| 94 |
|
simp2l |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
| 95 |
1 2 33 21 35 94
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 RngHom 𝑦 ) ) |
| 96 |
95
|
eleq2d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↔ 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) |
| 97 |
96
|
biimpd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) |
| 98 |
97
|
3exp |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) ) |
| 99 |
98
|
com14 |
⊢ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) ) |
| 100 |
99
|
3ad2ant2 |
⊢ ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) ) |
| 101 |
100
|
com13 |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) ) ) |
| 102 |
101
|
3imp |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) ) |
| 103 |
102
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) |
| 104 |
|
rnghmco |
⊢ ( ( 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ∧ 𝑓 ∈ ( 𝑤 RngHom 𝑥 ) ) → ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑤 RngHom 𝑦 ) ) |
| 105 |
103 44 104
|
syl2anc |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑤 RngHom 𝑦 ) ) |
| 106 |
|
simp2l |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → 𝑦 ∈ 𝐵 ) |
| 107 |
106
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑦 ∈ 𝐵 ) |
| 108 |
1 2 25 26 29 32 107 44 103
|
rngccoALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( 𝑔 ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑓 ) = ( 𝑔 ∘ 𝑓 ) ) |
| 109 |
1 2 25 21 29 107
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( 𝑤 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑤 RngHom 𝑦 ) ) |
| 110 |
105 108 109
|
3eltr4d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( 𝑔 ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑓 ) ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑦 ) ) |
| 111 |
|
coass |
⊢ ( ( ℎ ∘ 𝑔 ) ∘ 𝑓 ) = ( ℎ ∘ ( 𝑔 ∘ 𝑓 ) ) |
| 112 |
|
simp2r |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → 𝑧 ∈ 𝐵 ) |
| 113 |
112
|
adantl |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → 𝑧 ∈ 𝐵 ) |
| 114 |
|
simp2r |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → 𝑧 ∈ 𝐵 ) |
| 115 |
1 2 33 21 94 114
|
rngchomALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 RngHom 𝑧 ) ) |
| 116 |
115
|
eleq2d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) |
| 117 |
116
|
biimpd |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) |
| 118 |
117
|
3exp |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) ) ) |
| 119 |
118
|
com14 |
⊢ ( ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) ) ) |
| 120 |
119
|
3ad2ant3 |
⊢ ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑈 ∈ 𝑉 → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) ) ) |
| 121 |
120
|
com13 |
⊢ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑈 ∈ 𝑉 → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) ) ) |
| 122 |
121
|
3imp |
⊢ ( ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( 𝑈 ∈ 𝑉 → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) ) |
| 123 |
122
|
impcom |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ℎ ∈ ( 𝑦 RngHom 𝑧 ) ) |
| 124 |
|
rnghmco |
⊢ ( ( ℎ ∈ ( 𝑦 RngHom 𝑧 ) ∧ 𝑔 ∈ ( 𝑥 RngHom 𝑦 ) ) → ( ℎ ∘ 𝑔 ) ∈ ( 𝑥 RngHom 𝑧 ) ) |
| 125 |
123 103 124
|
syl2anc |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ℎ ∘ 𝑔 ) ∈ ( 𝑥 RngHom 𝑧 ) ) |
| 126 |
1 2 25 26 29 32 113 44 125
|
rngccoALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( ℎ ∘ 𝑔 ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ( ℎ ∘ 𝑔 ) ∘ 𝑓 ) ) |
| 127 |
1 2 25 26 29 107 113 105 123
|
rngccoALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ℎ ( ⟨ 𝑤 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) ( 𝑔 ∘ 𝑓 ) ) = ( ℎ ∘ ( 𝑔 ∘ 𝑓 ) ) ) |
| 128 |
111 126 127
|
3eqtr4a |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( ℎ ∘ 𝑔 ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑤 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) ( 𝑔 ∘ 𝑓 ) ) ) |
| 129 |
1 2 25 26 32 107 113 103 123
|
rngccoALTV |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) = ( ℎ ∘ 𝑔 ) ) |
| 130 |
129
|
oveq1d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ( ℎ ∘ 𝑔 ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) |
| 131 |
108
|
oveq2d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ℎ ( ⟨ 𝑤 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑓 ) ) = ( ℎ ( ⟨ 𝑤 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) ( 𝑔 ∘ 𝑓 ) ) ) |
| 132 |
128 130 131
|
3eqtr4d |
⊢ ( ( 𝑈 ∈ 𝑉 ∧ ( ( 𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝑓 ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑥 ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∧ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) ) → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑤 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑤 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑓 ) ) ) |
| 133 |
3 4 5 7 8 24 63 93 110 132
|
iscatd2 |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( I ↾ ( Base ‘ 𝑥 ) ) ) ) ) |