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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 21154 | . 2 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))) |
8 | oveq 7165 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗)) | |
9 | fveq1 6672 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦‘𝑗) = (𝑌‘𝑗)) | |
10 | 8, 9 | oveqan12d 7178 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗))) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦‘𝑗)) = ((𝑖𝑋𝑗) · (𝑌‘𝑗))) |
12 | 11 | mpteq2dv 5165 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) |
13 | 12 | oveq2d 7175 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) |
14 | 13 | mpteq2dv 5165 | . 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 6990 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ V) |
18 | 7, 14, 15, 16, 17 | ovmpod 7305 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 〈cop 4576 ↦ cmpt 5149 × cxp 5556 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Fincfn 8512 Basecbs 16486 .rcmulr 16569 Σg cgsu 16717 maVecMul cmvmul 21152 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-mvmul 21153 |
This theorem is referenced by: mvmulfv 21156 mavmulval 21157 |
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