1elcpmat - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 1elcpmat		Structured version Visualization version GIF version

Theorem 1elcpmat 22438

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 2731	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2731	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3	1, 2	ringidcl 20155	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
4	3	ancli 548	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
5	4	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
6	5	ad2antrl 725	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) → (𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅)))
7		eqid 2731	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
8		cpmatsrngpmat.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
9		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
10		eqid 2731	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
11	1, 7, 8, 9, 10	cply1coe0 22044	. . . . . . 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 2733	. . . . . . . 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 3176	. . . . . . 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 480	. . . . . 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 257	. . . . 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 20156	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
21	20	ancli 548	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
22	21	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Ring ∧ (0_g‘𝑅) ∈ (Base‘𝑅)))
23	1, 7, 8, 9, 10	cply1coe0 22044	. . . . . . . 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 725	. . . . . 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 480	. . . . . . . . . 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 2733	. . . . . . 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 3176	. . . . . 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 257	. . . . 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 809	. . . 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 3199	. . 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 764	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
37		simplr 766	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
38		simprl 768	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39		simprr 770	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
40		eqid 2731	. . . . . . . . 9 ⊢ (1_r‘𝐶) = (1_r‘𝐶)
41	8, 35, 10, 7, 2, 36, 37, 38, 39, 40	pmat1ovscd 22423	. . . . . . . 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 2733	. . . . 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 3176	. . . 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 3215	. . 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 257	. 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 22415	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
49		eqid 2731	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
50	49, 40	ringidcl 20155	. . . 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 22434	. . 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 1461	. 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 257	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ifcif 4528 ‘cfv 6543 (class class class)co 7412 Fincfn 8942 ℕcn 12217 Basecbs 17149 0_gc0g 17390 1_rcur 20076 Ringcrg 20128 algSccascl 21627 Poly₁cpl1 21921 coe₁cco1 21922 Mat cmat 22128 ConstPolyMat ccpmat 22426

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-ofr 7674 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-dsmm 21507 df-frlm 21522 df-ascl 21630 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-psr1 21924 df-vr1 21925 df-ply1 21926 df-coe1 21927 df-mamu 22107 df-mat 22129 df-cpmat 22429

This theorem is referenced by: cpmatsubgpmat 22443 cpmatsrgpmat 22444

W3C validator