Theorem madurid 22529

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 2729	. . 3 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2		eqid 2729	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2729	. . 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 22297	. . . . 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 22307	. . . 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 22526	. . . . . 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 7019	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵)
17	5, 2, 6	matbas2i 22307	. . . 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 22278	. 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 22323	. . . . 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 2782	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = · )
24	23	oveqd 7366	. 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 8776	. . . . . . . . . 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 7521	. . . . . 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 22528	. . . . 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 4482	. . . . . . . . 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 6653	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 Fn (𝑁 × 𝑁))
40		fnov 7480	. . . . . . . . . . . 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 2027	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → 𝑎 = 𝑑)
44	43	oveq1d 7364	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → (𝑎𝑀𝑐) = (𝑑𝑀𝑐))
45	44	ifeq1da 4508	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)))
46		ifid 4517	. . . . . . . . . . . . 13 ⊢ if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
47	45, 46	eqtrdi 2780	. . . . . . . . . . . 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 7428	. . . . . . . . . 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 2767	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
51	50	fveq2d 6826	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘𝑀) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
52	38, 51	eqtr2d 2765	. . . . . . 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 2729	. . . . . . . 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 7521	. . . . . . . 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 7520	. . . . . . . . 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 2933	. . . . . . . . . 10 ⊢ (¬ 𝑎 = 𝑏 → 𝑎 ≠ 𝑏)
68	67	necomd 2980	. . . . . . . . 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 22494	. . . . . . 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 4517	. . . . . . . . . . 11 ⊢ if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
72		oveq1 7356	. . . . . . . . . . . . 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 4508	. . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑀𝑐) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
76	75	ifeq2d 4497	. . . . . . . . 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 7428	. . . . . . . 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 6826	. . . . . . 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 4485	. . . . . . . 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 2774	. . . . . 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 2764	. . . 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 7426	. . 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 7360	. . . 4 ⊢ ((𝐷‘𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙ (1r‘𝐴))
87		crngring 20130	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
88	87	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
89	25, 5, 6, 2	mdetf 22480	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
90	89	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅))
91	90, 15	ffvelcdmd 7019	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘𝑀) ∈ (Base‘𝑅))
92		madurid.s	. . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐴)
93	5, 2, 92, 54	matsc 22335	. . . . 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 2776	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ 1 ) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
96	84, 95	eqtr4d 2767	. 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 2772	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436  ifcif 4476  ⟨cotp 4585   ↦ cmpt 5173   × cxp 5617   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753  Fincfn 8872  Basecbs 17120  ._rcmulr 17162   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  1_rcur 20066  Ringcrg 20118  CRingccrg 20119   maMul cmmul 22275   Mat cmat 22292   maDet cmdat 22469   maAdju cmadu 22517

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-addf 11088  ax-mulf 11089

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-word 14421  df-lsw 14470  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-reverse 14665  df-s2 14755  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-efmnd 18743  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-gim 19138  df-cntz 19196  df-oppg 19225  df-symg 19249  df-pmtr 19321  df-psgn 19370  df-evpm 19371  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-drng 20616  df-lmod 20765  df-lss 20835  df-sra 21077  df-rgmod 21078  df-cnfld 21262  df-zring 21354  df-zrh 21410  df-dsmm 21639  df-frlm 21654  df-mamu 22276  df-mat 22293  df-mdet 22470  df-madu 22519

This theorem is referenced by:  madulid  22530  matinv  22562  cpmadurid  22752  cpmidgsum2  22764