| Step | Hyp | Ref
| Expression |
| 1 | | mamucl.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 2 | | mamudir.p |
. . . . . 6
⊢ + =
(+g‘𝑅) |
| 3 | | mamucl.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 4 | | ringcmn 20279 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
| 5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ CMnd) |
| 6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑅 ∈ CMnd) |
| 7 | | mamudi.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ Fin) |
| 8 | 7 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑁 ∈ Fin) |
| 9 | 3 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
| 10 | | mamudir.x |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
| 11 | | elmapi 8889 |
. . . . . . . . . 10
⊢ (𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)) → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
| 12 | 10, 11 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
| 13 | 12 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑀 × 𝑁)⟶𝐵) |
| 14 | | simplrl 777 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑀) |
| 15 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) |
| 16 | 13, 14, 15 | fovcdmd 7605 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
| 17 | | mamudir.y |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 18 | | elmapi 8889 |
. . . . . . . . . 10
⊢ (𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)) → 𝑌:(𝑁 × 𝑂)⟶𝐵) |
| 19 | 17, 18 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌:(𝑁 × 𝑂)⟶𝐵) |
| 20 | 19 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌:(𝑁 × 𝑂)⟶𝐵) |
| 21 | | simplrr 778 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑂) |
| 22 | 20, 15, 21 | fovcdmd 7605 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑌𝑘) ∈ 𝐵) |
| 23 | | eqid 2737 |
. . . . . . . 8
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 24 | 1, 23 | ringcl 20247 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑗𝑌𝑘) ∈ 𝐵) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
| 25 | 9, 16, 22, 24 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) ∈ 𝐵) |
| 26 | | mamudir.z |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 27 | | elmapi 8889 |
. . . . . . . . . 10
⊢ (𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)) → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
| 29 | 28 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑍:(𝑁 × 𝑂)⟶𝐵) |
| 30 | 29, 15, 21 | fovcdmd 7605 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑍𝑘) ∈ 𝐵) |
| 31 | 1, 23 | ringcl 20247 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑗𝑍𝑘) ∈ 𝐵) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)) ∈ 𝐵) |
| 32 | 9, 16, 30, 31 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)) ∈ 𝐵) |
| 33 | | eqid 2737 |
. . . . . 6
⊢ (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) |
| 34 | | eqid 2737 |
. . . . . 6
⊢ (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))) |
| 35 | 1, 2, 6, 8, 25, 32, 33, 34 | gsummptfidmadd2 19944 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg ((𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) ∘f + (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))))) = ((𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) + (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))))) |
| 36 | 20 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌 Fn (𝑁 × 𝑂)) |
| 37 | 29 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑍 Fn (𝑁 × 𝑂)) |
| 38 | | mamudi.o |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑂 ∈ Fin) |
| 39 | | xpfi 9358 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ 𝑂 ∈ Fin) → (𝑁 × 𝑂) ∈ Fin) |
| 40 | 7, 38, 39 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 × 𝑂) ∈ Fin) |
| 41 | 40 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑁 × 𝑂) ∈ Fin) |
| 42 | | opelxpi 5722 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑂) → 〈𝑗, 𝑘〉 ∈ (𝑁 × 𝑂)) |
| 43 | 42 | ancoms 458 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ 𝑂 ∧ 𝑗 ∈ 𝑁) → 〈𝑗, 𝑘〉 ∈ (𝑁 × 𝑂)) |
| 44 | 43 | adantll 714 |
. . . . . . . . . . . . 13
⊢ (((𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂) ∧ 𝑗 ∈ 𝑁) → 〈𝑗, 𝑘〉 ∈ (𝑁 × 𝑂)) |
| 45 | 44 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 〈𝑗, 𝑘〉 ∈ (𝑁 × 𝑂)) |
| 46 | | fnfvof 7714 |
. . . . . . . . . . . 12
⊢ (((𝑌 Fn (𝑁 × 𝑂) ∧ 𝑍 Fn (𝑁 × 𝑂)) ∧ ((𝑁 × 𝑂) ∈ Fin ∧ 〈𝑗, 𝑘〉 ∈ (𝑁 × 𝑂))) → ((𝑌 ∘f + 𝑍)‘〈𝑗, 𝑘〉) = ((𝑌‘〈𝑗, 𝑘〉) + (𝑍‘〈𝑗, 𝑘〉))) |
| 47 | 36, 37, 41, 45, 46 | syl22anc 839 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑌 ∘f + 𝑍)‘〈𝑗, 𝑘〉) = ((𝑌‘〈𝑗, 𝑘〉) + (𝑍‘〈𝑗, 𝑘〉))) |
| 48 | | df-ov 7434 |
. . . . . . . . . . 11
⊢ (𝑗(𝑌 ∘f + 𝑍)𝑘) = ((𝑌 ∘f + 𝑍)‘〈𝑗, 𝑘〉) |
| 49 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢ (𝑗𝑌𝑘) = (𝑌‘〈𝑗, 𝑘〉) |
| 50 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢ (𝑗𝑍𝑘) = (𝑍‘〈𝑗, 𝑘〉) |
| 51 | 49, 50 | oveq12i 7443 |
. . . . . . . . . . 11
⊢ ((𝑗𝑌𝑘) + (𝑗𝑍𝑘)) = ((𝑌‘〈𝑗, 𝑘〉) + (𝑍‘〈𝑗, 𝑘〉)) |
| 52 | 47, 48, 51 | 3eqtr4g 2802 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑗(𝑌 ∘f + 𝑍)𝑘) = ((𝑗𝑌𝑘) + (𝑗𝑍𝑘))) |
| 53 | 52 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘)) = ((𝑖𝑋𝑗)(.r‘𝑅)((𝑗𝑌𝑘) + (𝑗𝑍𝑘)))) |
| 54 | 1, 2, 23 | ringdi 20258 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ ((𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑗𝑌𝑘) ∈ 𝐵 ∧ (𝑗𝑍𝑘) ∈ 𝐵)) → ((𝑖𝑋𝑗)(.r‘𝑅)((𝑗𝑌𝑘) + (𝑗𝑍𝑘))) = (((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) + ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))) |
| 55 | 9, 16, 22, 30, 54 | syl13anc 1374 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)((𝑗𝑌𝑘) + (𝑗𝑍𝑘))) = (((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) + ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))) |
| 56 | 53, 55 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘)) = (((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) + ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))) |
| 57 | 56 | mpteq2dva 5242 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘))) = (𝑗 ∈ 𝑁 ↦ (((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) + ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))))) |
| 58 | | eqidd 2738 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) |
| 59 | | eqidd 2738 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))) |
| 60 | 8, 25, 32, 58, 59 | offval2 7717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → ((𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) ∘f + (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))) = (𝑗 ∈ 𝑁 ↦ (((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)) + ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))))) |
| 61 | 57, 60 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘))) = ((𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) ∘f + (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))))) |
| 62 | 61 | oveq2d 7447 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘)))) = (𝑅 Σg ((𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))) ∘f + (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))))) |
| 63 | | mamudi.f |
. . . . . . 7
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) |
| 64 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑅 ∈ Ring) |
| 65 | | mamudi.m |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ Fin) |
| 66 | 65 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑀 ∈ Fin) |
| 67 | 38 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑂 ∈ Fin) |
| 68 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
| 69 | 17 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 70 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑖 ∈ 𝑀) |
| 71 | | simprr 773 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑘 ∈ 𝑂) |
| 72 | 63, 1, 23, 64, 66, 8, 67, 68, 69, 70, 71 | mamufv 22398 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑋𝐹𝑌)𝑘) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘))))) |
| 73 | 26 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 74 | 63, 1, 23, 64, 66, 8, 67, 68, 73, 70, 71 | mamufv 22398 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑋𝐹𝑍)𝑘) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘))))) |
| 75 | 72, 74 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → ((𝑖(𝑋𝐹𝑌)𝑘) + (𝑖(𝑋𝐹𝑍)𝑘)) = ((𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝑘)))) + (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗𝑍𝑘)))))) |
| 76 | 35, 62, 75 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘)))) = ((𝑖(𝑋𝐹𝑌)𝑘) + (𝑖(𝑋𝐹𝑍)𝑘))) |
| 77 | | ringmnd 20240 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) |
| 78 | 3, 77 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Mnd) |
| 79 | 1, 2 | mndvcl 18810 |
. . . . . . 7
⊢ ((𝑅 ∈ Mnd ∧ 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)) ∧ 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) → (𝑌 ∘f + 𝑍) ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 80 | 78, 17, 26, 79 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑌 ∘f + 𝑍) ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 81 | 80 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑌 ∘f + 𝑍) ∈ (𝐵 ↑m (𝑁 × 𝑂))) |
| 82 | 63, 1, 23, 64, 66, 8, 67, 68, 81, 70, 71 | mamufv 22398 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑋𝐹(𝑌 ∘f + 𝑍))𝑘) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗)(.r‘𝑅)(𝑗(𝑌 ∘f + 𝑍)𝑘))))) |
| 83 | 1, 3, 63, 65, 7, 38, 10, 17 | mamucl 22405 |
. . . . . . . 8
⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
| 84 | | elmapi 8889 |
. . . . . . . 8
⊢ ((𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → (𝑋𝐹𝑌):(𝑀 × 𝑂)⟶𝐵) |
| 85 | | ffn 6736 |
. . . . . . . 8
⊢ ((𝑋𝐹𝑌):(𝑀 × 𝑂)⟶𝐵 → (𝑋𝐹𝑌) Fn (𝑀 × 𝑂)) |
| 86 | 83, 84, 85 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑋𝐹𝑌) Fn (𝑀 × 𝑂)) |
| 87 | 86 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑋𝐹𝑌) Fn (𝑀 × 𝑂)) |
| 88 | 1, 3, 63, 65, 7, 38, 10, 26 | mamucl 22405 |
. . . . . . . 8
⊢ (𝜑 → (𝑋𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
| 89 | | elmapi 8889 |
. . . . . . . 8
⊢ ((𝑋𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → (𝑋𝐹𝑍):(𝑀 × 𝑂)⟶𝐵) |
| 90 | | ffn 6736 |
. . . . . . . 8
⊢ ((𝑋𝐹𝑍):(𝑀 × 𝑂)⟶𝐵 → (𝑋𝐹𝑍) Fn (𝑀 × 𝑂)) |
| 91 | 88, 89, 90 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑋𝐹𝑍) Fn (𝑀 × 𝑂)) |
| 92 | 91 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑋𝐹𝑍) Fn (𝑀 × 𝑂)) |
| 93 | | xpfi 9358 |
. . . . . . . 8
⊢ ((𝑀 ∈ Fin ∧ 𝑂 ∈ Fin) → (𝑀 × 𝑂) ∈ Fin) |
| 94 | 65, 38, 93 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝑀 × 𝑂) ∈ Fin) |
| 95 | 94 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑀 × 𝑂) ∈ Fin) |
| 96 | | opelxpi 5722 |
. . . . . . 7
⊢ ((𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂) → 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) |
| 97 | 96 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂)) |
| 98 | | fnfvof 7714 |
. . . . . 6
⊢ ((((𝑋𝐹𝑌) Fn (𝑀 × 𝑂) ∧ (𝑋𝐹𝑍) Fn (𝑀 × 𝑂)) ∧ ((𝑀 × 𝑂) ∈ Fin ∧ 〈𝑖, 𝑘〉 ∈ (𝑀 × 𝑂))) → (((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))‘〈𝑖, 𝑘〉) = (((𝑋𝐹𝑌)‘〈𝑖, 𝑘〉) + ((𝑋𝐹𝑍)‘〈𝑖, 𝑘〉))) |
| 99 | 87, 92, 95, 97, 98 | syl22anc 839 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))‘〈𝑖, 𝑘〉) = (((𝑋𝐹𝑌)‘〈𝑖, 𝑘〉) + ((𝑋𝐹𝑍)‘〈𝑖, 𝑘〉))) |
| 100 | | df-ov 7434 |
. . . . 5
⊢ (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘) = (((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))‘〈𝑖, 𝑘〉) |
| 101 | | df-ov 7434 |
. . . . . 6
⊢ (𝑖(𝑋𝐹𝑌)𝑘) = ((𝑋𝐹𝑌)‘〈𝑖, 𝑘〉) |
| 102 | | df-ov 7434 |
. . . . . 6
⊢ (𝑖(𝑋𝐹𝑍)𝑘) = ((𝑋𝐹𝑍)‘〈𝑖, 𝑘〉) |
| 103 | 101, 102 | oveq12i 7443 |
. . . . 5
⊢ ((𝑖(𝑋𝐹𝑌)𝑘) + (𝑖(𝑋𝐹𝑍)𝑘)) = (((𝑋𝐹𝑌)‘〈𝑖, 𝑘〉) + ((𝑋𝐹𝑍)‘〈𝑖, 𝑘〉)) |
| 104 | 99, 100, 103 | 3eqtr4g 2802 |
. . . 4
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘) = ((𝑖(𝑋𝐹𝑌)𝑘) + (𝑖(𝑋𝐹𝑍)𝑘))) |
| 105 | 76, 82, 104 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑋𝐹(𝑌 ∘f + 𝑍))𝑘) = (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘)) |
| 106 | 105 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖(𝑋𝐹(𝑌 ∘f + 𝑍))𝑘) = (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘)) |
| 107 | 1, 3, 63, 65, 7, 38, 10, 80 | mamucl 22405 |
. . . 4
⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
| 108 | | elmapi 8889 |
. . . 4
⊢ ((𝑋𝐹(𝑌 ∘f + 𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → (𝑋𝐹(𝑌 ∘f + 𝑍)):(𝑀 × 𝑂)⟶𝐵) |
| 109 | | ffn 6736 |
. . . 4
⊢ ((𝑋𝐹(𝑌 ∘f + 𝑍)):(𝑀 × 𝑂)⟶𝐵 → (𝑋𝐹(𝑌 ∘f + 𝑍)) Fn (𝑀 × 𝑂)) |
| 110 | 107, 108,
109 | 3syl 18 |
. . 3
⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) Fn (𝑀 × 𝑂)) |
| 111 | 1, 2 | mndvcl 18810 |
. . . . 5
⊢ ((𝑅 ∈ Mnd ∧ (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑂)) ∧ (𝑋𝐹𝑍) ∈ (𝐵 ↑m (𝑀 × 𝑂))) → ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
| 112 | 78, 83, 88, 111 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂))) |
| 113 | | elmapi 8889 |
. . . 4
⊢ (((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) ∈ (𝐵 ↑m (𝑀 × 𝑂)) → ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵) |
| 114 | | ffn 6736 |
. . . 4
⊢ (((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵 → ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) Fn (𝑀 × 𝑂)) |
| 115 | 112, 113,
114 | 3syl 18 |
. . 3
⊢ (𝜑 → ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) Fn (𝑀 × 𝑂)) |
| 116 | | eqfnov2 7563 |
. . 3
⊢ (((𝑋𝐹(𝑌 ∘f + 𝑍)) Fn (𝑀 × 𝑂) ∧ ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) Fn (𝑀 × 𝑂)) → ((𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖(𝑋𝐹(𝑌 ∘f + 𝑍))𝑘) = (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘))) |
| 117 | 110, 115,
116 | syl2anc 584 |
. 2
⊢ (𝜑 → ((𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖(𝑋𝐹(𝑌 ∘f + 𝑍))𝑘) = (𝑖((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))𝑘))) |
| 118 | 106, 117 | mpbird 257 |
1
⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))) |