Step | Hyp | Ref
| Expression |
1 | | natixp.2 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉)) |
2 | | natrcl.1 |
. . . . 5
⊢ 𝑁 = (𝐶 Nat 𝐷) |
3 | | natixp.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐶) |
4 | | nati.h |
. . . . 5
⊢ 𝐻 = (Hom ‘𝐶) |
5 | | eqid 2738 |
. . . . 5
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
6 | | nati.o |
. . . . 5
⊢ · =
(comp‘𝐷) |
7 | 2 | natrcl 17325 |
. . . . . . . 8
⊢ (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) |
8 | 1, 7 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) |
9 | 8 | simpld 498 |
. . . . . 6
⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
10 | | df-br 5031 |
. . . . . 6
⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
11 | 9, 10 | sylibr 237 |
. . . . 5
⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
12 | 8 | simprd 499 |
. . . . . 6
⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) |
13 | | df-br 5031 |
. . . . . 6
⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) |
14 | 12, 13 | sylibr 237 |
. . . . 5
⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) |
15 | 2, 3, 4, 5, 6, 11,
14 | isnat 17322 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥))))) |
16 | 1, 15 | mpbid 235 |
. . 3
⊢ (𝜑 → (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)))) |
17 | 16 | simprd 499 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥))) |
18 | | nati.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
19 | | nati.y |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
20 | 19 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) |
21 | | nati.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) |
22 | 21 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌)) |
23 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) |
24 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) |
25 | 23, 24 | oveq12d 7188 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
26 | 22, 25 | eleqtrrd 2836 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦)) |
27 | | simpllr 776 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑥 = 𝑋) |
28 | 27 | fveq2d 6678 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
29 | | simplr 769 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑦 = 𝑌) |
30 | 29 | fveq2d 6678 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑦) = (𝐹‘𝑌)) |
31 | 28, 30 | opeq12d 4769 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 = 〈(𝐹‘𝑋), (𝐹‘𝑌)〉) |
32 | 29 | fveq2d 6678 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑦) = (𝐾‘𝑌)) |
33 | 31, 32 | oveq12d 7188 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦)) = (〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))) |
34 | 29 | fveq2d 6678 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑦) = (𝐴‘𝑌)) |
35 | 27, 29 | oveq12d 7188 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌)) |
36 | | simpr 488 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑓 = 𝑅) |
37 | 35, 36 | fveq12d 6681 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅)) |
38 | 33, 34, 37 | oveq123d 7191 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅))) |
39 | 27 | fveq2d 6678 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑥) = (𝐾‘𝑋)) |
40 | 28, 39 | opeq12d 4769 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 〈(𝐹‘𝑥), (𝐾‘𝑥)〉 = 〈(𝐹‘𝑋), (𝐾‘𝑋)〉) |
41 | 40, 32 | oveq12d 7188 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦)) = (〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))) |
42 | 27, 29 | oveq12d 7188 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌)) |
43 | 42, 36 | fveq12d 6681 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅)) |
44 | 27 | fveq2d 6678 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑥) = (𝐴‘𝑋)) |
45 | 41, 43, 44 | oveq123d 7191 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋))) |
46 | 38, 45 | eqeq12d 2754 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)) ↔ ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋)))) |
47 | 26, 46 | rspcdv 3518 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋)))) |
48 | 20, 47 | rspcimdv 3516 |
. . 3
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋)))) |
49 | 18, 48 | rspcimdv 3516 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉 · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉 · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋)))) |
50 | 17, 49 | mpd 15 |
1
⊢ (𝜑 → ((𝐴‘𝑌)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉 · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(〈(𝐹‘𝑋), (𝐾‘𝑋)〉 · (𝐾‘𝑌))(𝐴‘𝑋))) |