![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cpm2mvalel | Structured version Visualization version GIF version |
Description: A (matrix) element of the result of an inverse matrix transformation. (Contributed by AV, 14-Dec-2019.) |
Ref | Expression |
---|---|
cpm2mfval.i | ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) |
cpm2mfval.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
Ref | Expression |
---|---|
cpm2mvalel | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝐼‘𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpm2mfval.i | . . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) | |
2 | cpm2mfval.s | . . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
3 | 1, 2 | cpm2mval 22115 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) |
4 | 3 | adantr 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) |
5 | oveq12 7367 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝑀𝑦) = (𝑋𝑀𝑌)) | |
6 | 5 | fveq2d 6847 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑋𝑀𝑌))) |
7 | 6 | fveq1d 6845 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0)) |
8 | 7 | adantl 483 | . 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑋𝑀𝑌))‘0)) |
9 | simprl 770 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑋 ∈ 𝑁) | |
10 | simprr 772 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → 𝑌 ∈ 𝑁) | |
11 | fvexd 6858 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → ((coe1‘(𝑋𝑀𝑌))‘0) ∈ V) | |
12 | 4, 8, 9, 10, 11 | ovmpod 7508 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝐼‘𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 Fincfn 8886 0cc0 11056 coe1cco1 21565 ConstPolyMat ccpmat 22068 cPolyMatToMat ccpmat2mat 22070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-cpmat2mat 22073 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |