Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mamuval | Structured version Visualization version GIF version |
Description: 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))) |
Ref | Expression |
---|---|
mamuval | ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (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 | 1, 2, 3, 4, 5, 6, 7 | mamufval 20924 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))))) |
9 | oveq 7151 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗)) | |
10 | oveq 7151 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑗𝑦𝑘) = (𝑗𝑌𝑘)) | |
11 | 9, 10 | oveqan12d 7164 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) |
13 | 12 | mpteq2dv 5153 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))) |
14 | 13 | oveq2d 7161 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) |
15 | 14 | mpoeq3dv 7222 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) |
16 | mamuval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) | |
17 | mamuval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) | |
18 | mpoexga 7764 | . . 3 ⊢ ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V) | |
19 | 5, 7, 18 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V) |
20 | 8, 15, 16, 17, 19 | ovmpod 7291 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cotp 4565 ↦ cmpt 5137 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 Fincfn 8497 Basecbs 16471 .rcmulr 16554 Σg cgsu 16702 maMul cmmul 20922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-mamu 20923 |
This theorem is referenced by: mamufv 20926 mamures 20929 mamucl 20938 mpomatmul 20983 mamutpos 20995 mat1dimmul 21013 dmatmul 21034 madurid 21181 cramerimplem2 21221 mat2pmatmul 21267 decpmatmul 21308 |
Copyright terms: Public domain | W3C validator |