![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22564 | . 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 6906 | . . . . . . 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 5250 | . . . 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 5250 | . 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 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 Basecbs 17245 .rcmulr 17299 Σg cgsu 17487 maVecMul cmvmul 22562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-mvmul 22563 |
This theorem is referenced by: mvmulfv 22566 mavmulval 22567 |
Copyright terms: Public domain | W3C validator |