Step | Hyp | Ref
| Expression |
1 | | estrccat.c |
. . 3
⊢ 𝐶 = (ExtStrCat‘𝑈) |
2 | | id 22 |
. . 3
⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉) |
3 | 1, 2 | estrcbas 16588 |
. 2
⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝐶)) |
4 | | eqidd 2611 |
. 2
⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶)) |
5 | | eqidd 2611 |
. 2
⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶)) |
6 | | fvex 6113 |
. . . 4
⊢
(ExtStrCat‘𝑈)
∈ V |
7 | 1, 6 | eqeltri 2684 |
. . 3
⊢ 𝐶 ∈ V |
8 | 7 | a1i 11 |
. 2
⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
9 | | biid 250 |
. 2
⊢ (((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) |
10 | | f1oi 6086 |
. . . 4
⊢ ( I
↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥) |
11 | | f1of 6050 |
. . . 4
⊢ (( I
↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)) |
12 | 10, 11 | mp1i 13 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)) |
13 | | simpl 472 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑉) |
14 | | eqid 2610 |
. . . 4
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
15 | | simpr 476 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) |
16 | | eqid 2610 |
. . . 4
⊢
(Base‘𝑥) =
(Base‘𝑥) |
17 | 1, 13, 14, 15, 15, 16, 16 | elestrchom 16591 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → (( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))) |
18 | 12, 17 | mpbird 246 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥)) |
19 | | simpl 472 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉) |
20 | | eqid 2610 |
. . . 4
⊢
(comp‘𝐶) =
(comp‘𝐶) |
21 | | simpr1l 1111 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝑈) |
22 | | simpr1r 1112 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝑈) |
23 | | eqid 2610 |
. . . 4
⊢
(Base‘𝑤) =
(Base‘𝑤) |
24 | | simpr31 1144 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥)) |
25 | 1, 19, 14, 21, 22, 23, 16 | elestrchom 16591 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓:(Base‘𝑤)⟶(Base‘𝑥))) |
26 | 24, 25 | mpbid 221 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) |
27 | 10, 11 | mp1i 13 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)) |
28 | 1, 19, 20, 21, 22, 22, 23, 16, 16, 26, 27 | estrcco 16593 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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)) |
29 | | fcoi2 5992 |
. . . 4
⊢ (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓) |
30 | 26, 29 | syl 17 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓) |
31 | 28, 30 | eqtrd 2644 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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) |
32 | | simpr2l 1113 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝑈) |
33 | | eqid 2610 |
. . . 4
⊢
(Base‘𝑦) =
(Base‘𝑦) |
34 | | simpr32 1145 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
35 | 1, 19, 14, 22, 32, 16, 33 | elestrchom 16591 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔:(Base‘𝑥)⟶(Base‘𝑦))) |
36 | 34, 35 | mpbid 221 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) |
37 | 1, 19, 20, 22, 22, 32, 16, 16, 33, 27, 36 | estrcco 16593 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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 ∘ ( I ↾ (Base‘𝑥)))) |
38 | | fcoi1 5991 |
. . . 4
⊢ (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔) |
39 | 36, 38 | syl 17 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔) |
40 | 37, 39 | eqtrd 2644 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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) |
41 | 1, 19, 20, 21, 22, 32, 23, 16, 33, 26, 36 | estrcco 16593 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(〈𝑤, 𝑥〉(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘ 𝑓)) |
42 | | fco 5971 |
. . . . 5
⊢ ((𝑔:(Base‘𝑥)⟶(Base‘𝑦) ∧ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) → (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦)) |
43 | 36, 26, 42 | syl2anc 691 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦)) |
44 | 1, 19, 14, 21, 32, 23, 33 | elestrchom 16591 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦) ↔ (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦))) |
45 | 43, 44 | mpbird 246 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦)) |
46 | 41, 45 | eqeltrd 2688 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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)) |
47 | | coass 5571 |
. . . 4
⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓)) |
48 | | simpr2r 1114 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈) |
49 | | eqid 2610 |
. . . . 5
⊢
(Base‘𝑧) =
(Base‘𝑧) |
50 | | simpr33 1146 |
. . . . . . 7
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) |
51 | 1, 19, 14, 32, 48, 33, 49 | elestrchom 16591 |
. . . . . . 7
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ℎ:(Base‘𝑦)⟶(Base‘𝑧))) |
52 | 50, 51 | mpbid 221 |
. . . . . 6
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ:(Base‘𝑦)⟶(Base‘𝑧)) |
53 | | fco 5971 |
. . . . . 6
⊢ ((ℎ:(Base‘𝑦)⟶(Base‘𝑧) ∧ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) → (ℎ ∘ 𝑔):(Base‘𝑥)⟶(Base‘𝑧)) |
54 | 52, 36, 53 | syl2anc 691 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘ 𝑔):(Base‘𝑥)⟶(Base‘𝑧)) |
55 | 1, 19, 20, 21, 22, 48, 23, 16, 49, 26, 54 | estrcco 16593 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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)) |
56 | 1, 19, 20, 21, 32, 48, 23, 33, 49, 43, 52 | estrcco 16593 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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))) |
57 | 47, 55, 56 | 3eqtr4a 2670 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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))) |
58 | 1, 19, 20, 22, 32, 48, 16, 33, 49, 36, 52 | estrcco 16593 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑔) = (ℎ ∘ 𝑔)) |
59 | 58 | oveq1d 6564 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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)) |
60 | 41 | oveq2d 6565 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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))) |
61 | 57, 59, 60 | 3eqtr4d 2654 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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))) |
62 | 3, 4, 5, 8, 9, 18,
31, 40, 46, 61 | iscatd2 16165 |
1
⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥))))) |