Theorem madurid 22663

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 2740	. . 3 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2		eqid 2740	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2740	. . 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 22429	. . . . 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 22441	. . . 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 22660	. . . . . 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 7114	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵)
17	5, 2, 6	matbas2i 22441	. . . 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 22410	. 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 22457	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
21	8, 20	sylan 579	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r‘𝐴))
22		madurid.t	. . . 4 ⊢ · = (.r‘𝐴)
23	21, 22	eqtr4di 2798	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = · )
24	23	oveqd 7460	. 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 1197	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑀 ∈ 𝐵)
27		simp1r 1198	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ CRing)
28		elmapi 8901	. . . . . . . . . 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 1192	. . . . . . 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 7616	. . . . . 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 22662	. . . . 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 4554	. . . . . . . . 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 6743	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 Fn (𝑁 × 𝑁))
40		fnov 7575	. . . . . . . . . . . 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 7458	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → (𝑎𝑀𝑐) = (𝑑𝑀𝑐))
45	44	ifeq1da 4579	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)))
46		ifid 4588	. . . . . . . . . . . . 13 ⊢ if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
47	45, 46	eqtrdi 2796	. . . . . . . . . . . 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 7523	. . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
51	50	fveq2d 6919	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘𝑀) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
52	38, 51	eqtr2d 2781	. . . . . . 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 1185	. . . . . 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 2740	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
55		simpl1r 1225	. . . . . . . 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 725	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
59		simpll2 1213	. . . . . . . . 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 7616	. . . . . . . 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 7615	. . . . . . . . 9 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
64	63	3impb 1115	. . . . . . . 8 ⊢ (((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
65		simpl3 1193	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ 𝑁)
66		simpl2 1192	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ 𝑁)
67		neqne 2954	. . . . . . . . . 10 ⊢ (¬ 𝑎 = 𝑏 → 𝑎 ≠ 𝑏)
68	67	necomd 3002	. . . . . . . . 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 22628	. . . . . . 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 4588	. . . . . . . . . . 11 ⊢ if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
72		oveq1 7450	. . . . . . . . . . . . 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 4579	. . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑀𝑐) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
76	75	ifeq2d 4568	. . . . . . . . 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 7523	. . . . . . . 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 6919	. . . . . . 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 4557	. . . . . . . 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 2790	. . . . . 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 2780	. . . 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 7521	. . 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 7454	. . . 4 ⊢ ((𝐷‘𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙ (1r‘𝐴))
87		crngring 20266	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
88	87	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
89	25, 5, 6, 2	mdetf 22614	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
90	89	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅))
91	90, 15	ffvelcdmd 7114	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘𝑀) ∈ (Base‘𝑅))
92		madurid.s	. . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐴)
93	5, 2, 92, 54	matsc 22469	. . . . 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 1371	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ (1r‘𝐴)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
95	86, 94	eqtrid 2792	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ 1 ) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))))
96	84, 95	eqtr4d 2783	. 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 2788	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488  ifcif 4548  ⟨cotp 4656   ↦ cmpt 5249   × cxp 5693   Fn wfn 6563  ⟶wf 6564  ‘cfv 6568  (class class class)co 7443   ∈ cmpo 7445   ↑_m cmap 8878  Fincfn 8997  Basecbs 17252  ._rcmulr 17306   ·_𝑠 cvsca 17309  0_gc0g 17493   Σ_g cgsu 17494  1_rcur 20202  Ringcrg 20254  CRingccrg 20255   maMul cmmul 22407   Mat cmat 22424   maDet cmdat 22603   maAdju cmadu 22651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764  ax-cnex 11234  ax-resscn 11235  ax-1cn 11236  ax-icn 11237  ax-addcl 11238  ax-addrcl 11239  ax-mulcl 11240  ax-mulrcl 11241  ax-mulcom 11242  ax-addass 11243  ax-mulass 11244  ax-distr 11245  ax-i2m1 11246  ax-1ne0 11247  ax-1rid 11248  ax-rnegex 11249  ax-rrecex 11250  ax-cnre 11251  ax-pre-lttri 11252  ax-pre-lttrn 11253  ax-pre-ltadd 11254  ax-pre-mulgt0 11255  ax-addf 11257  ax-mulf 11258

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5650  df-se 5651  df-we 5652  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-pred 6327  df-ord 6393  df-on 6394  df-lim 6395  df-suc 6396  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-isom 6577  df-riota 7399  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7708  df-om 7898  df-1st 8024  df-2nd 8025  df-supp 8196  df-tpos 8261  df-frecs 8316  df-wrecs 8347  df-recs 8421  df-rdg 8460  df-1o 8516  df-2o 8517  df-er 8757  df-map 8880  df-pm 8881  df-ixp 8950  df-en 8998  df-dom 8999  df-sdom 9000  df-fin 9001  df-fsupp 9426  df-sup 9505  df-oi 9573  df-card 10002  df-pnf 11320  df-mnf 11321  df-xr 11322  df-ltxr 11323  df-le 11324  df-sub 11516  df-neg 11517  df-div 11942  df-nn 12288  df-2 12350  df-3 12351  df-4 12352  df-5 12353  df-6 12354  df-7 12355  df-8 12356  df-9 12357  df-n0 12548  df-xnn0 12620  df-z 12634  df-dec 12753  df-uz 12898  df-rp 13052  df-fz 13562  df-fzo 13706  df-seq 14047  df-exp 14107  df-hash 14374  df-word 14557  df-lsw 14605  df-concat 14613  df-s1 14638  df-substr 14683  df-pfx 14713  df-splice 14792  df-reverse 14801  df-s2 14891  df-struct 17188  df-sets 17205  df-slot 17223  df-ndx 17235  df-base 17253  df-ress 17282  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17495  df-gsum 17496  df-prds 17501  df-pws 17503  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18672  df-sgrp 18751  df-mnd 18767  df-mhm 18812  df-submnd 18813  df-efmnd 18898  df-grp 18970  df-minusg 18971  df-sbg 18972  df-mulg 19102  df-subg 19157  df-ghm 19247  df-gim 19293  df-cntz 19351  df-oppg 19380  df-symg 19405  df-pmtr 19478  df-psgn 19527  df-evpm 19528  df-cmn 19818  df-abl 19819  df-mgp 20156  df-rng 20174  df-ur 20203  df-ring 20256  df-cring 20257  df-oppr 20354  df-dvdsr 20377  df-unit 20378  df-invr 20408  df-dvr 20421  df-rhm 20492  df-subrng 20566  df-subrg 20591  df-drng 20747  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-dsmm 21769  df-frlm 21784  df-mamu 22408  df-mat 22425  df-mdet 22604  df-madu 22653

This theorem is referenced by:  madulid  22664  matinv  22696  cpmadurid  22886  cpmidgsum2  22898