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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 20167 | . 2 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))) |
8 | oveq 6555 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗)) | |
9 | fveq1 6102 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑗) = (𝑌‘𝑗)) | |
10 | 8, 9 | oveqan12d 6568 | . . . . . 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 4673 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) |
13 | 12 | oveq2d 6565 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) |
14 | 13 | mpteq2dv 4673 | . 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 | mptexg 6389 | . . 3 ⊢ (𝑀 ∈ Fin → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V) | |
18 | 5, 17 | syl 17 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V) |
19 | 7, 14, 15, 16, 18 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 Basecbs 15695 .rcmulr 15769 Σg cgsu 15924 maVecMul cmvmul 20165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-mvmul 20166 |
This theorem is referenced by: mvmulfv 20169 mavmulval 20170 |
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