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| Mirrors > Home > MPE Home > Th. List > mvmulval | Structured version Visualization version GIF version | ||
| Description: Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.) |
| Ref | Expression |
|---|---|
| mvmulfval.x | ⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) |
| mvmulfval.b | ⊢ 𝐵 = (Base‘𝑅) |
| mvmulfval.t | ⊢ · = (.r‘𝑅) |
| mvmulfval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
| mvmulfval.m | ⊢ (𝜑 → 𝑀 ∈ Fin) |
| mvmulfval.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
| mvmulval.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
| mvmulval.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁)) |
| Ref | Expression |
|---|---|
| mvmulval | ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mvmulfval.x | . . 3 ⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) | |
| 2 | mvmulfval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | mvmulfval.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 4 | mvmulfval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
| 5 | mvmulfval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
| 6 | mvmulfval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
| 7 | 1, 2, 3, 4, 5, 6 | mvmulfval 22548 | . 2 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))) |
| 8 | oveq 7437 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗)) | |
| 9 | fveq1 6905 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑗) = (𝑌‘𝑗)) | |
| 10 | 8, 9 | oveqan12d 7450 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗))) |
| 11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗))) |
| 12 | 11 | mpteq2dv 5244 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) |
| 13 | 12 | oveq2d 7447 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) |
| 14 | 13 | mpteq2dv 5244 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
| 15 | mvmulval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) | |
| 16 | mvmulval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁)) | |
| 17 | 5 | mptexd 7244 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V) |
| 18 | 7, 14, 15, 16, 17 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 Basecbs 17247 .rcmulr 17298 Σg cgsu 17485 maVecMul cmvmul 22546 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-mvmul 22547 |
| This theorem is referenced by: mvmulfv 22550 mavmulval 22551 |
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