1elcpmat - Metamath Proof Explorer

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Theorem 1elcpmat 22216

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 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3	1, 2	ringidcl 20082	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
4	3	ancli 549	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
5	4	adantl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
6	5	ad2antrl 726	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
7		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
8		cpmatsrngpmat.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
9		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
10		eqid 2732	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
11	1, 7, 8, 9, 10	cply1coe0 21822	. . . . . . 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 4534	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))) = ((algSc‘𝑃)‘(1_r‘𝑅)))
14	13	fveq2d 6895	. . . . . . . . . 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 6893	. . . . . . . . 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 2734	. . . . . . . 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 3177	. . . . . . 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 481	. . . . . 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 20083	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
21	20	ancli 549	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
22	21	adantl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
23	1, 7, 8, 9, 10	cply1coe0 21822	. . . . . . . 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 726	. . . . . 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 4537	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 𝑗 → if(𝑖 = 𝑗, ((algSc‘𝑃)‘(1_r‘𝑅)), ((algSc‘𝑃)‘(0_g‘𝑅))) = ((algSc‘𝑃)‘(0_g‘𝑅)))
27	26	adantr 481	. . . . . . . . . 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 6895	. . . . . . . . 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 6893	. . . . . . . 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 2734	. . . . . . 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 3177	. . . . . 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 810	. . . 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 3200	. . 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 765	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37		simplr 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
38		simprl 769	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39		simprr 771	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
40		eqid 2732	. . . . . . . . 9 ⊢ (1_r‘𝐶) = (1_r‘𝐶)
41	8, 35, 10, 7, 2, 36, 37, 38, 39, 40	pmat1ovscd 22201	. . . . . . . 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 6895	. . . . . . 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 6893	. . . . . 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 2734	. . . . 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 3177	. . . 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 3216	. . 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 22193	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
49		eqid 2732	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
50	49, 40	ringidcl 20082	. . . 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 22212	. . 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 1462	. 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ifcif 4528 ‘cfv 6543 (class class class)co 7408 Fincfn 8938 ℕcn 12211 Basecbs 17143 0_gc0g 17384 1_rcur 20003 Ringcrg 20055 algSccascl 21406 Poly₁cpl1 21700 coe₁cco1 21701 Mat cmat 21906 ConstPolyMat ccpmat 22204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-subrg 20316 df-lmod 20472 df-lss 20542 df-sra 20784 df-rgmod 20785 df-dsmm 21286 df-frlm 21301 df-ascl 21409 df-psr 21461 df-mvr 21462 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-vr1 21704 df-ply1 21705 df-coe1 21706 df-mamu 21885 df-mat 21907 df-cpmat 22207

This theorem is referenced by: cpmatsubgpmat 22221 cpmatsrgpmat 22222

W3C validator