Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20192 | . 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) |
11 | oveq1 6657 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) | |
12 | oveq2 6658 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾)) | |
13 | 11, 12 | oveqan12d 6669 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 = 𝐾) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))) |
14 | 13 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))) |
15 | 14 | mpteq2dv 4745 | . . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) |
16 | 15 | oveq2d 6666 | . 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 6680 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V) | |
20 | 10, 16, 17, 18, 19 | ovmpt2d 6788 | 1 ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cotp 4185 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Fincfn 7955 Basecbs 15857 .rcmulr 15942 Σg cgsu 16101 maMul cmmul 20189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-mamu 20190 |
This theorem is referenced by: mamuass 20208 mamudi 20209 mamudir 20210 mamuvs1 20211 mamuvs2 20212 mamulid 20247 mamurid 20248 matmulcell 20251 mavmulass 20355 mvmumamul1 20360 mdetmul 20429 decpmatmullem 20576 matunitlindflem2 33406 |
Copyright terms: Public domain | W3C validator |