Step | Hyp | Ref
| Expression |
1 | | mamucl.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
2 | | eqid 2740 |
. . . . . 6
⊢
(0g‘𝑅) = (0g‘𝑅) |
3 | | eqid 2740 |
. . . . . 6
⊢
(+g‘𝑅) = (+g‘𝑅) |
4 | | mamuvs1.t |
. . . . . 6
⊢ · =
(.r‘𝑅) |
5 | | mamucl.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) |
6 | 5 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑅 ∈ Ring) |
7 | | mamudi.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ Fin) |
8 | 7 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑁 ∈ Fin) |
9 | | mamuvs1.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
10 | 9 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑋 ∈ 𝐵) |
11 | 5 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
12 | | mamuvs1.y |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
13 | | elmapi 8620 |
. . . . . . . . . 10
⊢ (𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁)) → 𝑌:(𝑀 × 𝑁)⟶𝐵) |
14 | 12, 13 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌:(𝑀 × 𝑁)⟶𝐵) |
15 | 14 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌:(𝑀 × 𝑁)⟶𝐵) |
16 | | simplrl 774 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑀) |
17 | | simpr 485 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) |
18 | 15, 16, 17 | fovrnd 7438 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑌𝑗) ∈ 𝐵) |
19 | | mamuvs1.z |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
20 | | elmapi 8620 |
. . . . . . . . . 10
⊢ (𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)) → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
21 | 19, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
22 | 21 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
23 | | simplrr 775 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑂) |
24 | 22, 17, 23 | fovrnd 7438 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑍𝑘) ∈ 𝐵) |
25 | 1, 4 | ringcl 19798 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑌𝑗) ∈ 𝐵 ∧ (𝑗𝑍𝑘) ∈ 𝐵) → ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)) ∈ 𝐵) |
26 | 11, 18, 24, 25 | syl3anc 1370 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)) ∈ 𝐵) |
27 | | eqid 2740 |
. . . . . . 7
⊢ (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))) |
28 | | ovexd 7306 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)) ∈ V) |
29 | | fvexd 6786 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (0g‘𝑅) ∈ V) |
30 | 27, 8, 28, 29 | fsuppmptdm 9117 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))) finSupp (0g‘𝑅)) |
31 | 1, 2, 3, 4, 6, 8, 10, 26, 30 | gsummulc2 19844 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))) = (𝑋 · (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))))) |
32 | | df-ov 7274 |
. . . . . . . . . 10
⊢ (𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) = ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)‘〈𝑖, 𝑗〉) |
33 | | simprl 768 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑖 ∈ 𝑀) |
34 | | opelxpi 5627 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑁) → 〈𝑖, 𝑗〉 ∈ (𝑀 × 𝑁)) |
35 | 33, 34 | sylan 580 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 〈𝑖, 𝑗〉 ∈ (𝑀 × 𝑁)) |
36 | | mamudi.m |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ Fin) |
37 | | xpfi 9063 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑀 × 𝑁) ∈ Fin) |
38 | 36, 7, 37 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀 × 𝑁) ∈ Fin) |
39 | 38 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑀 × 𝑁) ∈ Fin) |
40 | 9 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐵) |
41 | | ffn 6598 |
. . . . . . . . . . . . . 14
⊢ (𝑌:(𝑀 × 𝑁)⟶𝐵 → 𝑌 Fn (𝑀 × 𝑁)) |
42 | 12, 13, 41 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑌 Fn (𝑀 × 𝑁)) |
43 | 42 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌 Fn (𝑀 × 𝑁)) |
44 | | df-ov 7274 |
. . . . . . . . . . . . . 14
⊢ (𝑖𝑌𝑗) = (𝑌‘〈𝑖, 𝑗〉) |
45 | 44 | eqcomi 2749 |
. . . . . . . . . . . . 13
⊢ (𝑌‘〈𝑖, 𝑗〉) = (𝑖𝑌𝑗) |
46 | 45 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) ∧ 〈𝑖, 𝑗〉 ∈ (𝑀 × 𝑁)) → (𝑌‘〈𝑖, 𝑗〉) = (𝑖𝑌𝑗)) |
47 | 39, 40, 43, 46 | ofc1 7553 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) ∧ 〈𝑖, 𝑗〉 ∈ (𝑀 × 𝑁)) → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)‘〈𝑖, 𝑗〉) = (𝑋 · (𝑖𝑌𝑗))) |
48 | 35, 47 | mpdan 684 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)‘〈𝑖, 𝑗〉) = (𝑋 · (𝑖𝑌𝑗))) |
49 | 32, 48 | eqtrid 2792 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) = (𝑋 · (𝑖𝑌𝑗))) |
50 | 49 | oveq1d 7286 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘)) = ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘))) |
51 | 1, 4 | ringass 19801 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝑖𝑌𝑗) ∈ 𝐵 ∧ (𝑗𝑍𝑘) ∈ 𝐵)) → ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))) |
52 | 11, 40, 18, 24, 51 | syl13anc 1371 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))) |
53 | 50, 52 | eqtrd 2780 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))) |
54 | 53 | mpteq2dva 5179 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))) |
55 | 54 | oveq2d 7287 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))))) |
56 | | mamudi.f |
. . . . . . 7
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) |
57 | 36 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑀 ∈ Fin) |
58 | | mamudi.o |
. . . . . . . 8
⊢ (𝜑 → 𝑂 ∈ Fin) |
59 | 58 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑂 ∈ Fin) |
60 | 12 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
61 | 19 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
62 | | simprr 770 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑘 ∈ 𝑂) |
63 | 56, 1, 4, 6, 57, 8,
59, 60, 61, 33, 62 | mamufv 21534 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑌𝐹𝑍)𝑘) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))) |
64 | 63 | oveq2d 7287 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)) = (𝑋 · (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))))) |
65 | 31, 55, 64 | 3eqtr4d 2790 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘)))) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘))) |
66 | | fconst6g 6661 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐵 → ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵) |
67 | 9, 66 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵) |
68 | 1 | fvexi 6785 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
69 | | elmapg 8611 |
. . . . . . . . 9
⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑁) ∈ Fin) → (((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑁)) ↔ ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵)) |
70 | 68, 38, 69 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → (((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑁)) ↔ ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵)) |
71 | 67, 70 | mpbird 256 |
. . . . . . 7
⊢ (𝜑 → ((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
72 | 1, 4 | ringvcl 21545 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ ((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑁)) ∧ 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) → (((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
73 | 5, 71, 12, 72 | syl3anc 1370 |
. . . . . 6
⊢ (𝜑 → (((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
74 | 73 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
75 | 56, 1, 4, 6, 57, 8,
59, 74, 61, 33, 62 | mamufv 21534 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍)𝑘) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝑗) · (𝑗𝑍𝑘))))) |
76 | | df-ov 7274 |
. . . . 5
⊢ (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘) = ((((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))‘〈𝑖, 𝑘〉) |
77 | | opelxpi 5627 |
. . . . . . 7
⊢ ((𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂) → 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) |
78 | 77 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) |
79 | | xpfi 9063 |
. . . . . . . . 9
⊢ ((𝑀 ∈ Fin ∧ 𝑂 ∈ Fin) → (𝑀 × 𝑂) ∈ Fin) |
80 | 36, 58, 79 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 × 𝑂) ∈ Fin) |
81 | 80 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑀 × 𝑂) ∈ Fin) |
82 | 1, 5, 56, 36, 7, 58, 12, 19 | mamucl 21546 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
83 | | elmapi 8620 |
. . . . . . . . 9
⊢ ((𝑌𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → (𝑌𝐹𝑍):(𝑀 × 𝑂)⟶𝐵) |
84 | | ffn 6598 |
. . . . . . . . 9
⊢ ((𝑌𝐹𝑍):(𝑀 × 𝑂)⟶𝐵 → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂)) |
85 | 82, 83, 84 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂)) |
86 | 85 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂)) |
87 | | df-ov 7274 |
. . . . . . . . 9
⊢ (𝑖(𝑌𝐹𝑍)𝑘) = ((𝑌𝐹𝑍)‘〈𝑖, 𝑘〉) |
88 | 87 | eqcomi 2749 |
. . . . . . . 8
⊢ ((𝑌𝐹𝑍)‘〈𝑖, 𝑘〉) = (𝑖(𝑌𝐹𝑍)𝑘) |
89 | 88 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) → ((𝑌𝐹𝑍)‘〈𝑖, 𝑘〉) = (𝑖(𝑌𝐹𝑍)𝑘)) |
90 | 81, 10, 86, 89 | ofc1 7553 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) → ((((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))‘〈𝑖, 𝑘〉) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘))) |
91 | 78, 90 | mpdan 684 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → ((((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))‘〈𝑖, 𝑘〉) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘))) |
92 | 76, 91 | eqtrid 2792 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘))) |
93 | 65, 75, 92 | 3eqtr4d 2790 |
. . 3
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘)) |
94 | 93 | ralrimivva 3117 |
. 2
⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘)) |
95 | 1, 5, 56, 36, 7, 58, 73, 19 | mamucl 21546 |
. . . 4
⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
96 | | elmapi 8620 |
. . . 4
⊢
(((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍):(𝑀 × 𝑂)⟶𝐵) |
97 | | ffn 6598 |
. . . 4
⊢
(((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍):(𝑀 × 𝑂)⟶𝐵 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂)) |
98 | 95, 96, 97 | 3syl 18 |
. . 3
⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂)) |
99 | | fconst6g 6661 |
. . . . . . 7
⊢ (𝑋 ∈ 𝐵 → ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵) |
100 | 9, 99 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵) |
101 | | elmapg 8611 |
. . . . . . 7
⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑂) ∈ Fin) → (((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑂)) ↔ ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵)) |
102 | 68, 80, 101 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑂)) ↔ ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵)) |
103 | 100, 102 | mpbird 256 |
. . . . 5
⊢ (𝜑 → ((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
104 | 1, 4 | ringvcl 21545 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ ((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑m (𝑀 × 𝑂)) ∧ (𝑌𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂))) → (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
105 | 5, 103, 82, 104 | syl3anc 1370 |
. . . 4
⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
106 | | elmapi 8620 |
. . . 4
⊢ ((((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵) |
107 | | ffn 6598 |
. . . 4
⊢ ((((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵 → (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂)) |
108 | 105, 106,
107 | 3syl 18 |
. . 3
⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂)) |
109 | | eqfnov2 7398 |
. . 3
⊢
((((((𝑀 ×
𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂) ∧ (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂)) → (((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘))) |
110 | 98, 108, 109 | syl2anc 584 |
. 2
⊢ (𝜑 → (((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))𝑘))) |
111 | 94, 110 | mpbird 256 |
1
⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))) |