Theorem cramerimp 22692

Description: One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."): The ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)

Hypotheses

Ref	Expression
cramerimp.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cramerimp.b	⊢ 𝐵 = (Base‘𝐴)
cramerimp.v	⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
cramerimp.e	⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
cramerimp.h	⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
cramerimp.x	⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
cramerimp.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
cramerimp.q	⊢ / = (/r‘𝑅)

Assertion

Ref	Expression
cramerimp	⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = ((𝐷‘𝐻) / (𝐷‘𝑋)))

Proof of Theorem cramerimp

Step	Hyp	Ref	Expression
1		crngring 20242	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ Ring)
3	2	3ad2ant1 1134	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝑅 ∈ Ring)
4		cramerimp.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 maDet 𝑅)
5		cramerimp.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6		cramerimp.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
7		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	4, 5, 6, 7	mdetf 22601	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
9	8	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝐷:𝐵⟶(Base‘𝑅))
10	9	3ad2ant1 1134	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝐷:𝐵⟶(Base‘𝑅))
11		cramerimp.e	. . . . . 6 ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
12	5, 6	matrcl 22416	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
13	12	simpld 494	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin)
15	2, 14	anim12i 613	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
16	15	3adant3 1133	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
17		ne0i 4341	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑁 → 𝑁 ≠ ∅)
18	1, 17	anim12ci 614	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
19	18	anim1i 615	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)))
20	19	3adant3 1133	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)))
21		simpl 482	. . . . . . . . 9 ⊢ (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑋 · 𝑍) = 𝑌)
22	21	3ad2ant3 1136	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑋 · 𝑍) = 𝑌)
23		cramerimp.v	. . . . . . . . 9 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
24		cramerimp.x	. . . . . . . . 9 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
25	5, 6, 23, 24	slesolvec 22685	. . . . . . . 8 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
26	20, 22, 25	sylc 65	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝑍 ∈ 𝑉)
27		simpr 484	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝐼 ∈ 𝑁)
28	27	3ad2ant1 1134	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝐼 ∈ 𝑁)
29		eqid 2737	. . . . . . . 8 ⊢ (1r‘𝐴) = (1r‘𝐴)
30	5, 6, 23, 29	ma1repvcl 22576	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
31	16, 26, 28, 30	syl12anc 837	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
32	11, 31	eqeltrid 2845	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝐸 ∈ 𝐵)
33	10, 32	ffvelcdmd 7105	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝐷‘𝐸) ∈ (Base‘𝑅))
34		simpr 484	. . . . 5 ⊢ (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
35	34	3ad2ant3 1136	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
36		eqid 2737	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
37		cramerimp.q	. . . . 5 ⊢ / = (/r‘𝑅)
38		eqid 2737	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
39	7, 36, 37, 38	dvrcan3 20410	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐷‘𝐸) ∈ (Base‘𝑅) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (((𝐷‘𝐸)(.r‘𝑅)(𝐷‘𝑋)) / (𝐷‘𝑋)) = (𝐷‘𝐸))
40	3, 33, 35, 39	syl3anc 1373	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (((𝐷‘𝐸)(.r‘𝑅)(𝐷‘𝑋)) / (𝐷‘𝑋)) = (𝐷‘𝐸))
41		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ CRing)
42	41	3ad2ant1 1134	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝑅 ∈ CRing)
43	7, 36	unitcl 20375	. . . . . . 7 ⊢ ((𝐷‘𝑋) ∈ (Unit‘𝑅) → (𝐷‘𝑋) ∈ (Base‘𝑅))
44	43	adantl 481	. . . . . 6 ⊢ (((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝐷‘𝑋) ∈ (Base‘𝑅))
45	44	3ad2ant3 1136	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝐷‘𝑋) ∈ (Base‘𝑅))
46	7, 38	crngcom 20248	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐷‘𝐸) ∈ (Base‘𝑅) ∧ (𝐷‘𝑋) ∈ (Base‘𝑅)) → ((𝐷‘𝐸)(.r‘𝑅)(𝐷‘𝑋)) = ((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)))
47	42, 33, 45, 46	syl3anc 1373	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → ((𝐷‘𝐸)(.r‘𝑅)(𝐷‘𝑋)) = ((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)))
48	47	oveq1d 7446	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (((𝐷‘𝐸)(.r‘𝑅)(𝐷‘𝑋)) / (𝐷‘𝑋)) = (((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)) / (𝐷‘𝑋)))
49	14	adantl 481	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → 𝑁 ∈ Fin)
50	41	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → 𝑅 ∈ CRing)
51	27	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → 𝐼 ∈ 𝑁)
52	49, 50, 51	3jca 1129	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁))
53	52	3adant3 1133	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁))
54	5, 23, 11, 4	cramerimplem1 22689	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ 𝑍 ∈ 𝑉) → (𝐷‘𝐸) = (𝑍‘𝐼))
55	53, 26, 54	syl2anc 584	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝐷‘𝐸) = (𝑍‘𝐼))
56	40, 48, 55	3eqtr3rd 2786	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = (((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)) / (𝐷‘𝑋)))
57		cramerimp.h	. . . . 5 ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
58	5, 6, 23, 11, 57, 24, 4, 38	cramerimplem3 22691	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)) = (𝐷‘𝐻))
59	58	3adant3r 1182	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → ((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)) = (𝐷‘𝐻))
60	59	oveq1d 7446	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (((𝐷‘𝑋)(.r‘𝑅)(𝐷‘𝐸)) / (𝐷‘𝑋)) = ((𝐷‘𝐻) / (𝐷‘𝑋)))
61	56, 60	eqtrd 2777	1 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = ((𝐷‘𝐻) / (𝐷‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  ∅c0 4333  ⟨cop 4632  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  Fincfn 8985  Basecbs 17247  ._rcmulr 17298  1_rcur 20178  Ringcrg 20230  CRingccrg 20231  Unitcui 20355  /_rcdvr 20400   Mat cmat 22411   maVecMul cmvmul 22546   matRepV cmatrepV 22563   maDet cmdat 22590

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-addf 11234  ax-mulf 11235

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-reverse 14797  df-s2 14887  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-efmnd 18882  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-symg 19387  df-pmtr 19460  df-psgn 19509  df-evpm 19510  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-dsmm 21752  df-frlm 21767  df-mamu 22395  df-mat 22412  df-mvmul 22547  df-marrep 22564  df-marepv 22565  df-subma 22583  df-mdet 22591  df-minmar1 22641

This theorem is referenced by:  cramerlem1  22693