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Mirrors > Home > MPE Home > Th. List > mvmulfv | Structured version Visualization version GIF version |
Description: A cell/element in the vector resulting from a 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)) |
mvmulfv.i | ⊢ (𝜑 → 𝐼 ∈ 𝑀) |
Ref | Expression |
---|---|
mvmulfv | ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (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 | mvmulval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) | |
8 | mvmulval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mvmulval 22489 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | oveq1 7426 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) | |
11 | 10 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) |
12 | 11 | oveq1d 7434 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗))) |
13 | 12 | mpteq2dv 5251 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) |
14 | 13 | oveq2d 7435 | . 2 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
15 | mvmulfv.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑀) | |
16 | ovexd 7454 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V) | |
17 | 9, 14, 15, 16 | fvmptd 7011 | 1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 〈cop 4636 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 Fincfn 8964 Basecbs 17183 .rcmulr 17237 Σg cgsu 17425 maVecMul cmvmul 22486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-mvmul 22487 |
This theorem is referenced by: mvmumamul1 22500 |
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