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Theorem cpmatpmat 20922

Description: A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
cpmat.b	⊢ 𝐵 = (Base‘𝐶)

**Assertion**
Ref	Expression
cpmatpmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵)

Proof of Theorem cpmatpmat Dummy variables 𝑚 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

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 20921	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)})
6	5	eleq2d 2844	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)}))
7		elrabi 3566	. . 3 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)} → 𝑀 ∈ 𝐵)
8	6, 7	syl6bi 245	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵))
9	8	3impia 1106	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∀wral 3089 {crab 3093 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 ℕcn 11374 Basecbs 16255 0_gc0g 16486 Poly₁cpl1 19943 coe₁cco1 19944 Mat cmat 20617 ConstPolyMat ccpmat 20915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-cpmat 20918

This theorem is referenced by: cpmatelimp 20924 cpmatelimp2 20926 cpmatacl 20928 cpmatinvcl 20929 cpmatmcl 20931 cpm2mf 20964 m2cpminvid2lem 20966 m2cpminvid2 20967 m2cpmfo 20968 m2cpmrngiso 20970

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