Theorem madurid 22580

Description: Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)

Hypotheses

Ref	Expression
madurid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madurid.b	⊢ 𝐵 = (Base‘𝐴)
madurid.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
madurid.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
madurid.i	⊢ 1 = (1r‘𝐴)
madurid.t	⊢ · = (.r‘𝐴)
madurid.s	⊢ ∙ = ( ·𝑠 ‘𝐴)

Assertion

Ref	Expression
madurid	⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))

Proof of Theorem madurid

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2		eqid 2735	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2735	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		simpr 484	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
5		madurid.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6		madurid.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
7	5, 6	matrcl 22348	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
8	7	simpld 494	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
9	8	adantr 480	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑁 ∈ Fin)
10	5, 2, 6	matbas2i 22358	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
11	10	adantr 480	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
12		madurid.j	. . . . . . 7 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
13	5, 12, 6	maduf 22577	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
14	13	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐽:𝐵⟶𝐵)
15		simpl 482	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ 𝐵)
16	14, 15	ffvelcdmd 7074	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵)
17	5, 2, 6	matbas2i 22358	. . . 4 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
18	16, 17	syl 17	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
19	1, 2, 3, 4, 9, 9, 9, 11, 18	mamuval 22329	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝐽‘𝑀)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏))))))
20	5, 1	matmulr 22374	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
21	8, 20	sylan 580	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
22		madurid.t	. . . 4 ⊢ · = (.r‘𝐴)
23	21, 22	eqtr4di 2788	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = · )
24	23	oveqd 7420	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝐽‘𝑀)) = (𝑀 · (𝐽‘𝑀)))
25		madurid.d	. . . . . 6 ⊢ 𝐷 = (𝑁 maDet 𝑅)
26		simp1l 1198	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑀 ∈ 𝐵)
27		simp1r 1199	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ CRing)
28		elmapi 8861	. . . . . . . . . 10 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
29	11, 28	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
30	29	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
31	30	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
32		simpl2 1193	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑎 ∈ 𝑁)
33		simpr 484	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
34	31, 32, 33	fovcdmd 7577	. . . . . 6 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅))
35		simp3 1138	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
36	5, 12, 6, 25, 3, 2, 26, 27, 34, 35	madugsum 22579	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏)))) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
37		iftrue 4506	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (𝐷‘𝑀))
38	37	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (𝐷‘𝑀))
39	29	ffnd 6706	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 Fn (𝑁 × 𝑁))
40		fnov 7536	. . . . . . . . . . . 12 ⊢ (𝑀 Fn (𝑁 × 𝑁) ↔ 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐)))
41	39, 40	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐)))
42	41	adantr 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐)))
43		equtr2 2026	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → 𝑎 = 𝑑)
44	43	oveq1d 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → (𝑎𝑀𝑐) = (𝑑𝑀𝑐))
45	44	ifeq1da 4532	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)))
46		ifid 4541	. . . . . . . . . . . . 13 ⊢ if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
47	45, 46	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐))
48	47	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐))
49	48	mpoeq3dv 7484	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐)))
50	42, 49	eqtr4d 2773	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
51	50	fveq2d 6879	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘𝑀) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
52	38, 51	eqtr2d 2771	. . . . . . 7 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))
53	52	3ad2antl1 1186	. . . . . 6 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))
54		eqid 2735	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
55		simpl1r 1226	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑅 ∈ CRing)
56	9	3ad2ant1 1133	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑁 ∈ Fin)
57	56	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑁 ∈ Fin)
58	30	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
59		simpll2 1214	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑎 ∈ 𝑁)
60		simpr 484	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
61	58, 59, 60	fovcdmd 7577	. . . . . . . 8 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅))
62	30	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
63	62	fovcdmda 7576	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
64	63	3impb 1114	. . . . . . . 8 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
65		simpl3 1194	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ 𝑁)
66		simpl2 1193	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ 𝑁)
67		neqne 2940	. . . . . . . . . 10 ⊢ (¬ 𝑎 = 𝑏 → 𝑎 ≠ 𝑏)
68	67	necomd 2987	. . . . . . . . 9 ⊢ (¬ 𝑎 = 𝑏 → 𝑏 ≠ 𝑎)
69	68	adantl 481	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ≠ 𝑎)
70	25, 2, 54, 55, 57, 61, 64, 65, 66, 69	mdetralt2 22545	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) = (0g‘𝑅))
71		ifid 4541	. . . . . . . . . . 11 ⊢ if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
72		oveq1 7410	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑎 → (𝑑𝑀𝑐) = (𝑎𝑀𝑐))
73	72	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 = 𝑎) → (𝑑𝑀𝑐) = (𝑎𝑀𝑐))
74	73	ifeq1da 4532	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
75	71, 74	eqtr3id 2784	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑀𝑐) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
76	75	ifeq2d 4521	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
77	76	mpoeq3dv 7484	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
78	77	fveq2d 6879	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))))
79		iffalse 4509	. . . . . . . 8 ⊢ (¬ 𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (0g‘𝑅))
80	79	adantl 481	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (0g‘𝑅))
81	70, 78, 80	3eqtr4d 2780	. . . . . 6 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))
82	53, 81	pm2.61dan 812	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))
83	36, 82	eqtrd 2770	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))
84	83	mpoeq3dva 7482	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏))))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
85		madurid.i	. . . . 5 ⊢ 1 = (1r‘𝐴)
86	85	oveq2i 7414	. . . 4 ⊢ ((𝐷‘𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙ (1r‘𝐴))
87		crngring 20203	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
88	87	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
89	25, 5, 6, 2	mdetf 22531	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
90	89	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅))
91	90, 15	ffvelcdmd 7074	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘𝑀) ∈ (Base‘𝑅))
92		madurid.s	. . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐴)
93	5, 2, 92, 54	matsc 22386	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐷‘𝑀) ∈ (Base‘𝑅)) → ((𝐷‘𝑀) ∙ (1r‘𝐴)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
94	9, 88, 91, 93	syl3anc 1373	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ (1r‘𝐴)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
95	86, 94	eqtrid 2782	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ 1 ) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
96	84, 95	eqtr4d 2773	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏))))) = ((𝐷‘𝑀) ∙ 1 ))
97	19, 24, 96	3eqtr3d 2778	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ifcif 4500  ⟨cotp 4609   ↦ cmpt 5201   × cxp 5652   Fn wfn 6525  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403   ∈ cmpo 7405   ↑_m cmap 8838  Fincfn 8957  Basecbs 17226  ._rcmulr 17270   ·_𝑠 cvsca 17273  0_gc0g 17451   Σ_g cgsu 17452  1_rcur 20139  Ringcrg 20191  CRingccrg 20192   maMul cmmul 22326   Mat cmat 22343   maDet cmdat 22520   maAdju cmadu 22568

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-addf 11206  ax-mulf 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669  df-om 7860  df-1st 7986  df-2nd 7987  df-supp 8158  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-er 8717  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9372  df-sup 9452  df-oi 9522  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12851  df-rp 13007  df-fz 13523  df-fzo 13670  df-seq 14018  df-exp 14078  df-hash 14347  df-word 14530  df-lsw 14579  df-concat 14587  df-s1 14612  df-substr 14657  df-pfx 14687  df-splice 14766  df-reverse 14775  df-s2 14865  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-mulr 17283  df-starv 17284  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-unif 17292  df-hom 17293  df-cco 17294  df-0g 17453  df-gsum 17454  df-prds 17459  df-pws 17461  df-mre 17596  df-mrc 17597  df-acs 17599  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-mhm 18759  df-submnd 18760  df-efmnd 18845  df-grp 18917  df-minusg 18918  df-sbg 18919  df-mulg 19049  df-subg 19104  df-ghm 19194  df-gim 19240  df-cntz 19298  df-oppg 19327  df-symg 19349  df-pmtr 19421  df-psgn 19470  df-evpm 19471  df-cmn 19761  df-abl 19762  df-mgp 20099  df-rng 20111  df-ur 20140  df-ring 20193  df-cring 20194  df-oppr 20295  df-dvdsr 20315  df-unit 20316  df-invr 20346  df-dvr 20359  df-rhm 20430  df-subrng 20504  df-subrg 20528  df-drng 20689  df-lmod 20817  df-lss 20887  df-sra 21129  df-rgmod 21130  df-cnfld 21314  df-zring 21406  df-zrh 21462  df-dsmm 21690  df-frlm 21705  df-mamu 22327  df-mat 22344  df-mdet 22521  df-madu 22570

This theorem is referenced by:  madulid  22581  matinv  22613  cpmadurid  22803  cpmidgsum2  22815