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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 21245 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | 5 | eleq2d 2895 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)})) |
7 | elrabi 3672 | . . 3 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} → 𝑀 ∈ 𝐵) | |
8 | 6, 7 | syl6bi 254 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) |
9 | 8 | 3impia 1109 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℕcn 11626 Basecbs 16471 0gc0g 16701 Poly1cpl1 20273 coe1cco1 20274 Mat cmat 20944 ConstPolyMat ccpmat 21239 |
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-sep 5194 ax-nul 5201 ax-pr 5320 |
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-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-cpmat 21242 |
This theorem is referenced by: cpmatelimp 21248 cpmatelimp2 21250 cpmatacl 21252 cpmatinvcl 21253 cpmatmcl 21255 cpm2mf 21288 m2cpminvid2lem 21290 m2cpminvid2 21291 m2cpmfo 21292 m2cpmrngiso 21294 |
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