1elcpmat - Metamath Proof Explorer

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Theorem 1elcpmat 22437

Description: The identity of the ring of all polynomial matrices over the ring 𝑅 is a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.)

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

**Assertion**
Ref	Expression
1elcpmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ 𝑆)

Proof of Theorem 1elcpmat Dummy variables 𝑖 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2730	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3	1, 2	ringidcl 20154	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
4	3	ancli 547	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
5	4	adantl 480	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
6	5	ad2antrl 724	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
7		eqid 2730	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
8		cpmatsrngpmat.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
9		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
10		eqid 2730	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
11	1, 7, 8, 9, 10	cply1coe0 22043	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)) → ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛) = (0_g‘𝑅))
12	6, 11	syl 17	. . . . . 6 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛) = (0_g‘𝑅))
13		iftrue 4533	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))) = ((algSc‘𝑃)‘(1_r‘𝑅)))
14	13	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅)))) = (coe₁‘((algSc‘𝑃)‘(1_r‘𝑅))))
15	14	fveq1d 6892	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛))
16	15	eqeq1d 2732	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅) ↔ ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛) = (0_g‘𝑅)))
17	16	ralbidv 3175	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛) = (0_g‘𝑅)))
18	17	adantr 479	. . . . . 6 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(1_r‘𝑅)))‘𝑛) = (0_g‘𝑅)))
19	12, 18	mpbird 256	. . . . 5 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅))
20	1, 7	ring0cl 20155	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
21	20	ancli 547	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
22	21	adantl 480	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
23	1, 7, 8, 9, 10	cply1coe0 22043	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)) → ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛) = (0_g‘𝑅))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛) = (0_g‘𝑅))
25	24	ad2antrl 724	. . . . . 6 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛) = (0_g‘𝑅))
26		iffalse 4536	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 𝑗 → if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))) = ((algSc‘𝑃)‘(0_g‘𝑅)))
27	26	adantr 479	. . . . . . . . . 10 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))) = ((algSc‘𝑃)‘(0_g‘𝑅)))
28	27	fveq2d 6894	. . . . . . . . 9 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅)))) = (coe₁‘((algSc‘𝑃)‘(0_g‘𝑅))))
29	28	fveq1d 6892	. . . . . . . 8 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛))
30	29	eqeq1d 2732	. . . . . . 7 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅) ↔ ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛) = (0_g‘𝑅)))
31	30	ralbidv 3175	. . . . . 6 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑛) = (0_g‘𝑅)))
32	25, 31	mpbird 256	. . . . 5 ⊢ ((¬ 𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅))
33	19, 32	pm2.61ian 808	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅))
34	33	ralrimivva 3198	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅))
35		cpmatsrngpmat.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
36		simpll 763	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37		simplr 765	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
38		simprl 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39		simprr 769	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
40		eqid 2730	. . . . . . . . 9 ⊢ (1_r‘𝐶) = (1_r‘𝐶)
41	8, 35, 10, 7, 2, 36, 37, 38, 39, 40	pmat1ovscd 22422	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(1_r‘𝐶)𝑗) = if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))
42	41	fveq2d 6894	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe₁‘(𝑖(1_r‘𝐶)𝑗)) = (coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅)))))
43	42	fveq1d 6892	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛))
44	43	eqeq1d 2732	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅) ↔ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅)))
45	44	ralbidv 3175	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑛 ∈ ℕ ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅)))
46	45	2ralbidva 3214	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))))‘𝑛) = (0_g‘𝑅)))
47	34, 46	mpbird 256	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅))
48	8, 35	pmatring 22414	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
49		eqid 2730	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
50	49, 40	ringidcl 20154	. . . 4 ⊢ (𝐶 ∈ Ring → (1_r‘𝐶) ∈ (Base‘𝐶))
51	48, 50	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ (Base‘𝐶))
52		cpmatsrngpmat.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
53	52, 8, 35, 49	cpmatel 22433	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (1_r‘𝐶) ∈ (Base‘𝐶)) → ((1_r‘𝐶) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅)))
54	51, 53	mpd3an3 1460	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((1_r‘𝐶) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑛 ∈ ℕ ((coe₁‘(𝑖(1_r‘𝐶)𝑗))‘𝑛) = (0_g‘𝑅)))
55	47, 54	mpbird 256	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ifcif 4527 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 ℕcn 12216 Basecbs 17148 0_gc0g 17389 1_rcur 20075 Ringcrg 20127 algSccascl 21626 Poly₁cpl1 21920 coe₁cco1 21921 Mat cmat 22127 ConstPolyMat ccpmat 22425

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-ascl 21629 df-psr 21681 df-mvr 21682 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-vr1 21924 df-ply1 21925 df-coe1 21926 df-mamu 22106 df-mat 22128 df-cpmat 22428

This theorem is referenced by: cpmatsubgpmat 22442 cpmatsrgpmat 22443

W3C validator