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Mirrors > Home > MPE Home > Th. List > cpmatpmat | Structured version Visualization version GIF version |
Description: A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.) |
Ref | Expression |
---|---|
cpmat.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
cpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
cpmat.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
cpmatpmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmat.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | cpmat.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | cpmat.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
4 | cpmat.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmat 22211 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | 5 | eleq2d 2820 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)})) |
7 | elrabi 3678 | . . 3 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} → 𝑀 ∈ 𝐵) | |
8 | 6, 7 | syl6bi 253 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) |
9 | 8 | 3impia 1118 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℕcn 12212 Basecbs 17144 0gc0g 17385 Poly1cpl1 21701 coe1cco1 21702 Mat cmat 21907 ConstPolyMat ccpmat 22205 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-cpmat 22208 |
This theorem is referenced by: cpmatelimp 22214 cpmatelimp2 22216 cpmatacl 22218 cpmatinvcl 22219 cpmatmcl 22221 cpm2mf 22254 m2cpminvid2lem 22256 m2cpminvid2 22257 m2cpmfo 22258 m2cpmrngiso 22260 |
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