Step | Hyp | Ref
| Expression |
1 | | xpccofval.t |
. . . 4
⊢ 𝑇 = (𝐶 ×c 𝐷) |
2 | | eqid 2622 |
. . . 4
⊢
(Base‘𝐶) =
(Base‘𝐶) |
3 | | eqid 2622 |
. . . 4
⊢
(Base‘𝐷) =
(Base‘𝐷) |
4 | | eqid 2622 |
. . . 4
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
5 | | eqid 2622 |
. . . 4
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
6 | | xpccofval.o1 |
. . . 4
⊢ · =
(comp‘𝐶) |
7 | | xpccofval.o2 |
. . . 4
⊢ ∙ =
(comp‘𝐷) |
8 | | simpl 473 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V) |
9 | | simpr 477 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V) |
10 | | xpccofval.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑇) |
11 | 1, 2, 3 | xpcbas 16818 |
. . . . . 6
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘𝑇) |
12 | 10, 11 | eqtr4i 2647 |
. . . . 5
⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷)) |
13 | 12 | a1i 11 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷))) |
14 | | xpccofval.k |
. . . . . 6
⊢ 𝐾 = (Hom ‘𝑇) |
15 | 1, 10, 4, 5, 14 | xpchomfval 16819 |
. . . . 5
⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))) |
16 | 15 | a1i 11 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))) |
17 | | eqidd 2623 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13,
16, 17 | xpcval 16817 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {〈(Base‘ndx),
𝐵〉, 〈(Hom
‘ndx), 𝐾〉,
〈(comp‘ndx), (𝑥
∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))〉}) |
19 | | catstr 16617 |
. . 3
⊢
{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐾〉, 〈(comp‘ndx),
(𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} Struct 〈1, ;15〉 |
20 | | ccoid 16077 |
. . 3
⊢ comp =
Slot (comp‘ndx) |
21 | | snsstp3 4349 |
. . 3
⊢
{〈(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} ⊆
{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐾〉, 〈(comp‘ndx),
(𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} |
22 | | fvex 6201 |
. . . . . . 7
⊢
(Base‘𝑇)
∈ V |
23 | 10, 22 | eqeltri 2697 |
. . . . . 6
⊢ 𝐵 ∈ V |
24 | 23, 23 | xpex 6962 |
. . . . 5
⊢ (𝐵 × 𝐵) ∈ V |
25 | 24, 23 | mpt2ex 7247 |
. . . 4
⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) ∈ V |
26 | 25 | a1i 11 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) ∈ V) |
27 | | xpccofval.o |
. . 3
⊢ 𝑂 = (comp‘𝑇) |
28 | 18, 19, 20, 21, 26, 27 | strfv3 15908 |
. 2
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))) |
29 | | mpt20 6725 |
. . . 4
⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd
‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) = ∅ |
30 | 29 | eqcomi 2631 |
. . 3
⊢ ∅ =
(𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd
‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) |
31 | | fnxpc 16816 |
. . . . . . . 8
⊢
×c Fn (V × V) |
32 | | fndm 5990 |
. . . . . . . 8
⊢ (
×c Fn (V × V) → dom
×c = (V × V)) |
33 | 31, 32 | ax-mp 5 |
. . . . . . 7
⊢ dom
×c = (V × V) |
34 | 33 | ndmov 6818 |
. . . . . 6
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c
𝐷) =
∅) |
35 | 1, 34 | syl5eq 2668 |
. . . . 5
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅) |
36 | 35 | fveq2d 6195 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) →
(comp‘𝑇) =
(comp‘∅)) |
37 | 20 | str0 15911 |
. . . 4
⊢ ∅ =
(comp‘∅) |
38 | 36, 27, 37 | 3eqtr4g 2681 |
. . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = ∅) |
39 | 35 | fveq2d 6195 |
. . . . . . 7
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) →
(Base‘𝑇) =
(Base‘∅)) |
40 | | base0 15912 |
. . . . . . 7
⊢ ∅ =
(Base‘∅) |
41 | 39, 10, 40 | 3eqtr4g 2681 |
. . . . . 6
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅) |
42 | 41 | xpeq2d 5139 |
. . . . 5
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅)) |
43 | | xp0 5552 |
. . . . 5
⊢ (𝐵 × ∅) =
∅ |
44 | 42, 43 | syl6eq 2672 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐵) = ∅) |
45 | | eqidd 2623 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑔 ∈ ((2nd
‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉) = (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) |
46 | 44, 41, 45 | mpt2eq123dv 6717 |
. . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))) |
47 | 30, 38, 46 | 3eqtr4a 2682 |
. 2
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉))) |
48 | 28, 47 | pm2.61i 176 |
1
⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉 ·
(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉 ∙
(2nd ‘𝑦))(2nd ‘𝑓))〉)) |