Theorem madutpos 22867

Description: The adjuct of a transposed matrix is the transposition of the adjunct of the matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)

Hypotheses

Ref	Expression
maduf.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
maduf.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
maduf.b	⊢ 𝐵 = (Base‘𝐴)

Assertion

Ref	Expression
madutpos	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

Proof of Theorem madutpos

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

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))
2	1	tposmpo 8262	. . . . . . . 8 ⊢ tpos (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) = (𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))
3		orcom 884	. . . . . . . . . . 11 ⊢ ((𝑑 = 𝑎 ∨ 𝑐 = 𝑏) ↔ (𝑐 = 𝑏 ∨ 𝑑 = 𝑎))
4	3	a1i 11	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑑 = 𝑎 ∨ 𝑐 = 𝑏) ↔ (𝑐 = 𝑏 ∨ 𝑑 = 𝑎)))
5		ancom 466	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝑏 ∧ 𝑑 = 𝑎) ↔ (𝑑 = 𝑎 ∧ 𝑐 = 𝑏))
6	5	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑐 = 𝑏 ∧ 𝑑 = 𝑎) ↔ (𝑑 = 𝑎 ∧ 𝑐 = 𝑏)))
7	6	ifbid 4506	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)) = if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)))
8		ovtpos 8240	. . . . . . . . . . . 12 ⊢ (𝑐tpos 𝑀𝑑) = (𝑑𝑀𝑐)
9	8	eqcomi 2769	. . . . . . . . . . 11 ⊢ (𝑑𝑀𝑐) = (𝑐tpos 𝑀𝑑)
10	9	a1i 11	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑑𝑀𝑐) = (𝑐tpos 𝑀𝑑))
11	4, 7, 10	ifbieq12d 4511	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)) = if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑)))
12	11	mpoeq3dv 7493	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) = (𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑))))
13	2, 12	eqtrid 2807	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → tpos (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) = (𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑))))
14	13	fveq2d 6883	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑁 maDet 𝑅)‘tpos (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))) = ((𝑁 maDet 𝑅)‘(𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑)))))
15		simpll 779	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ CRing)
16		maduf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		maduf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
19	16, 18	matrcl 22637	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
20	19	simpld 500	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
21	20	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑁 ∈ Fin)
22		simp1ll 1255	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑅 ∈ CRing)
23		crngring 20387	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24		eqid 2760	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	17, 24	ringidcl 20409	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26		eqid 2760	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	17, 26	ring0cl 20411	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	25, 27	ifcld 4529	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
29	22, 23, 28	3syl 19	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
30	16, 17, 18	matbas2i 22647	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
31		elmapi 8851	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
32	30, 31	syl 18	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝐵 → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
33	32	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
34	33	fovcdmda 7586	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
35	34	3impb 1132	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
36	29, 35	ifcld 4529	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)) ∈ (Base‘𝑅))
37	16, 17, 18, 21, 15, 36	matbas2d 22648	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) ∈ 𝐵)
38		eqid 2760	. . . . . . . 8 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅)
39	38, 16, 18	mdettpos 22836	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐))) ∈ 𝐵) → ((𝑁 maDet 𝑅)‘tpos (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))) = ((𝑁 maDet 𝑅)‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))))
40	15, 37, 39	syl2anc 596	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑁 maDet 𝑅)‘tpos (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))) = ((𝑁 maDet 𝑅)‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))))
41	14, 40	eqtr3d 2797	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑁 maDet 𝑅)‘(𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑)))) = ((𝑁 maDet 𝑅)‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))))
42	16, 18	mattposcl 22678	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
43	42	adantl 487	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
44	43	adantr 486	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → tpos 𝑀 ∈ 𝐵)
45		simprl 783	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
46		simprr 785	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑏 ∈ 𝑁)
47		maduf.j	. . . . . . 7 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
48	16, 38, 47, 18, 24, 26	maducoeval2 22865	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ tpos 𝑀 ∈ 𝐵) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝐽‘tpos 𝑀)𝑏) = ((𝑁 maDet 𝑅)‘(𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑)))))
49	15, 44, 45, 46, 48	syl211anc 1403	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = ((𝑁 maDet 𝑅)‘(𝑐 ∈ 𝑁, 𝑑 ∈ 𝑁 ↦ if((𝑐 = 𝑏 ∨ 𝑑 = 𝑎), if((𝑑 = 𝑎 ∧ 𝑐 = 𝑏), (1r‘𝑅), (0g‘𝑅)), (𝑐tpos 𝑀𝑑)))))
50		simplr 781	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
51	16, 38, 47, 18, 24, 26	maducoeval2 22865	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑏 ∈ 𝑁 ∧ 𝑎 ∈ 𝑁) → (𝑏(𝐽‘𝑀)𝑎) = ((𝑁 maDet 𝑅)‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))))
52	15, 50, 46, 45, 51	syl211anc 1403	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑏(𝐽‘𝑀)𝑎) = ((𝑁 maDet 𝑅)‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑑 = 𝑎 ∨ 𝑐 = 𝑏), if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)), (𝑑𝑀𝑐)))))
53	41, 49, 52	3eqtr4d 2805	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎))
54		ovtpos 8240	. . . 4 ⊢ (𝑎tpos (𝐽‘𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎)
55	53, 54	eqtr4di 2813	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
56	55	ralrimivva 3205	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
57	16, 47, 18	maduf 22866	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
58	57	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:𝐵⟶𝐵)
59	58, 43	ffvelcdmd 7079	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ 𝐵)
60	16, 17, 18	matbas2i 22647	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ 𝐵 → (𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
61		elmapi 8851	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → (𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
62		ffn 6703	. . . 4 ⊢ ((𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
63	59, 60, 61, 62	4syl 20	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
64	57	ffvelcdmda 7078	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵)
65	16, 18	mattposcl 22678	. . . . 5 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ 𝐵)
66	16, 17, 18	matbas2i 22647	. . . . 5 ⊢ (tpos (𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
67	64, 65, 66	3syl 19	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
68		elmapi 8851	. . . 4 ⊢ (tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
69		ffn 6703	. . . 4 ⊢ (tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
70	67, 68, 69	3syl 19	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
71		eqfnov2 7544	. . 3 ⊢ (((𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁) ∧ tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁)) → ((𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏)))
72	63, 70, 71	syl2anc 596	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏)))
73	56, 72	mpbird 260	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ifcif 4482   × cxp 5653   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  tpos ctpos 8224   ↑_m cmap 8829  Fincfn 8955  Basecbs 17304  0_gc0g 17527  1_rcur 20323  Ringcrg 20375  CRingccrg 20376   Mat cmat 22632   maDet cmdat 22809   maAdju cmadu 22857

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  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 proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-xnn0 12605  df-z 12619  df-dec 12740  df-uz 12891  df-rp 13046  df-fz 13565  df-fzo 13713  df-seq 14069  df-exp 14129  df-hash 14398  df-word 14582  df-lsw 14631  df-concat 14639  df-s1 14666  df-substr 14712  df-pfx 14744  df-splice 14822  df-reverse 14831  df-s2 14922  df-struct 17242  df-sets 17259  df-slot 17277  df-ndx 17289  df-base 17305  df-ress 17326  df-plusg 17358  df-mulr 17359  df-starv 17360  df-sca 17361  df-vsca 17362  df-ip 17363  df-tset 17364  df-ple 17365  df-ds 17367  df-unif 17368  df-hom 17369  df-cco 17370  df-0g 17529  df-gsum 17530  df-prds 17535  df-pws 17537  df-mre 17673  df-mrc 17674  df-acs 17676  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-mhm 18894  df-submnd 18895  df-efmnd 18981  df-grp 19063  df-minusg 19064  df-mulg 19194  df-subg 19249  df-ghm 19344  df-gim 19389  df-cntz 19447  df-oppg 19476  df-symg 19500  df-pmtr 19572  df-psgn 19621  df-evpm 19622  df-cmn 19912  df-abl 19913  df-mgp 20277  df-rng 20291  df-ur 20324  df-ring 20377  df-cring 20378  df-oppr 20481  df-dvdsr 20501  df-unit 20502  df-invr 20532  df-dvr 20545  df-rhm 20616  df-subrng 20711  df-subrg 20735  df-drng 20895  df-sra 21360  df-rgmod 21361  df-cnfld 21589  df-zring 21663  df-zrh 21719  df-dsmm 21948  df-frlm 21963  df-mat 22633  df-mdet 22810  df-madu 22859

This theorem is used by:  madulid  22870