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Theorem cpmatpmat 22632

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 22631	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)})
6	5	eleq2d 2815	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)}))
7		elrabi 3678	. . 3 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)} → 𝑀 ∈ 𝐵)
8	6, 7	biimtrdi 252	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵))
9	8	3impia 1114	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {crab 3430 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 ℕcn 12250 Basecbs 17187 0_gc0g 17428 Poly₁cpl1 22103 coe₁cco1 22104 Mat cmat 22327 ConstPolyMat ccpmat 22625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-cpmat 22628

This theorem is referenced by: cpmatelimp 22634 cpmatelimp2 22636 cpmatacl 22638 cpmatinvcl 22639 cpmatmcl 22641 cpm2mf 22674 m2cpminvid2lem 22676 m2cpminvid2 22677 m2cpmfo 22678 m2cpmrngiso 22680

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