Theorem cramerimp 21437

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 19428	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 484	. . . . 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 2738	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	4, 5, 6, 7	mdetf 21346	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
9	8	adantr 484	. . . . . 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 21163	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
13	12	simpld 498	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
14	13	adantr 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin)
15	2, 14	anim12i 616	. . . . . . . 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 4223	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑁 → 𝑁 ≠ ∅)
18	1, 17	anim12ci 617	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
19	18	anim1i 618	. . . . . . . . 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 486	. . . . . . . . 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 21430	. . . . . . . 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 488	. . . . . . . 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 2738	. . . . . . . 8 ⊢ (1r‘𝐴) = (1r‘𝐴)
30	5, 6, 23, 29	ma1repvcl 21321	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
31	16, 26, 28, 30	syl12anc 836	. . . . . 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 2837	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → 𝐸 ∈ 𝐵)
33	10, 32	ffvelrnd 6862	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝐷‘𝐸) ∈ (Base‘𝑅))
34		simpr 488	. . . . 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 2738	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
37		cramerimp.q	. . . . 5 ⊢ / = (/r‘𝑅)
38		eqid 2738	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
39	7, 36, 37, 38	dvrcan3 19564	. . . 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 1372	. . 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 486	. . . . . 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 19531	. . . . . . 7 ⊢ ((𝐷‘𝑋) ∈ (Unit‘𝑅) → (𝐷‘𝑋) ∈ (Base‘𝑅))
44	43	adantl 485	. . . . . 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 19434	. . . . 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 1372	. . . 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 7185	. . 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 485	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → 𝑁 ∈ Fin)
50	41	adantr 484	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → 𝑅 ∈ CRing)
51	27	adantr 484	. . . . . 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 21434	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ 𝑍 ∈ 𝑉) → (𝐷‘𝐸) = (𝑍‘𝐼))
55	53, 26, 54	syl2anc 587	. . 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 2782	. 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 21436	. . . 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 7185	. 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 2773	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 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2934  Vcvv 3398  ∅c0 4211  ⟨cop 4522  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170   ↑_m cmap 8437  Fincfn 8555  Basecbs 16586  ._rcmulr 16669  1_rcur 19370  Ringcrg 19416  CRingccrg 19417  Unitcui 19511  /_rcdvr 19554   Mat cmat 21158   maVecMul cmvmul 21291   matRepV cmatrepV 21308   maDet cmdat 21335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-addf 10694  ax-mulf 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-xor 1507  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-ot 4525  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-tpos 7921  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-er 8320  df-map 8439  df-pm 8440  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-sup 8979  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-xnn0 12049  df-z 12063  df-dec 12180  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-seq 13461  df-exp 13522  df-hash 13783  df-word 13956  df-lsw 14004  df-concat 14012  df-s1 14039  df-substr 14092  df-pfx 14122  df-splice 14201  df-reverse 14210  df-s2 14299  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-starv 16683  df-sca 16684  df-vsca 16685  df-ip 16686  df-tset 16687  df-ple 16688  df-ds 16690  df-unif 16691  df-hom 16692  df-cco 16693  df-0g 16818  df-gsum 16819  df-prds 16824  df-pws 16826  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-submnd 18073  df-efmnd 18150  df-grp 18222  df-minusg 18223  df-sbg 18224  df-mulg 18343  df-subg 18394  df-ghm 18474  df-gim 18517  df-cntz 18565  df-oppg 18592  df-symg 18614  df-pmtr 18688  df-psgn 18737  df-evpm 18738  df-cmn 19026  df-abl 19027  df-mgp 19359  df-ur 19371  df-srg 19375  df-ring 19418  df-cring 19419  df-oppr 19495  df-dvdsr 19513  df-unit 19514  df-invr 19544  df-dvr 19555  df-rnghom 19589  df-drng 19623  df-subrg 19652  df-lmod 19755  df-lss 19823  df-sra 20063  df-rgmod 20064  df-cnfld 20218  df-zring 20290  df-zrh 20324  df-dsmm 20548  df-frlm 20563  df-mamu 21137  df-mat 21159  df-mvmul 21292  df-marrep 21309  df-marepv 21310  df-subma 21328  df-mdet 21336  df-minmar1 21386

This theorem is referenced by:  cramerlem1  21438