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Mirrors > Home > MPE Home > Th. List > mamufv | Structured version Visualization version GIF version |
Description: A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
Ref | Expression |
---|---|
mamufval.f | ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) |
mamufval.b | ⊢ 𝐵 = (Base‘𝑅) |
mamufval.t | ⊢ · = (.r‘𝑅) |
mamufval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
mamufval.m | ⊢ (𝜑 → 𝑀 ∈ Fin) |
mamufval.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mamufval.p | ⊢ (𝜑 → 𝑃 ∈ Fin) |
mamuval.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) |
mamuval.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) |
mamufv.i | ⊢ (𝜑 → 𝐼 ∈ 𝑀) |
mamufv.k | ⊢ (𝜑 → 𝐾 ∈ 𝑃) |
Ref | Expression |
---|---|
mamufv | ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mamufval.f | . . 3 ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) | |
2 | mamufval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | mamufval.t | . . 3 ⊢ · = (.r‘𝑅) | |
4 | mamufval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
5 | mamufval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
6 | mamufval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
7 | mamufval.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ Fin) | |
8 | mamuval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) | |
9 | mamuval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mamuval 20996 | . 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) |
11 | oveq1 7162 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) | |
12 | oveq2 7163 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾)) | |
13 | 11, 12 | oveqan12d 7174 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 = 𝐾) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))) |
14 | 13 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))) |
15 | 14 | mpteq2dv 5161 | . . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) |
16 | 15 | oveq2d 7171 | . 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) |
17 | mamufv.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑀) | |
18 | mamufv.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝑃) | |
19 | ovexd 7190 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V) | |
20 | 10, 16, 17, 18, 19 | ovmpod 7301 | 1 ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cotp 4574 ↦ cmpt 5145 × cxp 5552 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 Fincfn 8508 Basecbs 16482 .rcmulr 16565 Σg cgsu 16713 maMul cmmul 20993 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-ot 4575 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-mamu 20994 |
This theorem is referenced by: mamuass 21010 mamudi 21011 mamudir 21012 mamuvs1 21013 mamuvs2 21014 mamulid 21049 mamurid 21050 matmulcell 21053 mavmulass 21157 mvmumamul1 21162 mdetmul 21231 decpmatmullem 21378 matunitlindflem2 34888 |
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