Theorem madutpos 22494

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 2726	. . . . . . . . 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 8246	. . . . . . . 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 867	. . . . . . . . . . 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 460	. . . . . . . . . . . 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 4546	. . . . . . . . . 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 8224	. . . . . . . . . . . 12 ⊢ (𝑐tpos 𝑀𝑑) = (𝑑𝑀𝑐)
9	8	eqcomi 2735	. . . . . . . . . . 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 4551	. . . . . . . . 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 7483	. . . . . . . 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 2778	. . . . . . 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 6888	. . . . . 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 764	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ CRing)
16		maduf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		eqid 2726	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		maduf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
19	16, 18	matrcl 22262	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
20	19	simpld 494	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
21	20	ad2antlr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑁 ∈ Fin)
22		simp1ll 1233	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑅 ∈ CRing)
23		crngring 20147	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24		eqid 2726	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	17, 24	ringidcl 20162	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26		eqid 2726	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	17, 26	ring0cl 20163	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	25, 27	ifcld 4569	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → if((𝑐 = 𝑏 ∧ 𝑑 = 𝑎), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
29	22, 23, 28	3syl 18	. . . . . . . . 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 22274	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
31		elmapi 8842	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝐵 → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
33	32	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
34	33	fovcdmda 7574	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
35	34	3impb 1112	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
36	29, 35	ifcld 4569	. . . . . . . 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 22275	. . . . . . 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 2726	. . . . . . . 8 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅)
39	38, 16, 18	mdettpos 22463	. . . . . . 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 583	. . . . . 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 2768	. . . . 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 22305	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
44	43	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → tpos 𝑀 ∈ 𝐵)
45		simprl 768	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
46		simprr 770	. . . . . 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 22492	. . . . . 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 1373	. . . . 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 766	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
51	16, 38, 47, 18, 24, 26	maducoeval2 22492	. . . . . 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 1373	. . . . 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 2776	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎))
54		ovtpos 8224	. . . 4 ⊢ (𝑎tpos (𝐽‘𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎)
55	53, 54	eqtr4di 2784	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
56	55	ralrimivva 3194	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
57	16, 47, 18	maduf 22493	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
58	57	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:𝐵⟶𝐵)
59	58, 43	ffvelcdmd 7080	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ 𝐵)
60	16, 17, 18	matbas2i 22274	. . . . 5 ⊢ ((𝐽‘tpos 𝑀) ∈ 𝐵 → (𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
61	59, 60	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
62		elmapi 8842	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → (𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
63		ffn 6710	. . . 4 ⊢ ((𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
64	61, 62, 63	3syl 18	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
65	57	ffvelcdmda 7079	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵)
66	16, 18	mattposcl 22305	. . . . 5 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ 𝐵)
67	16, 17, 18	matbas2i 22274	. . . . 5 ⊢ (tpos (𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
68	65, 66, 67	3syl 18	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
69		elmapi 8842	. . . 4 ⊢ (tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
70		ffn 6710	. . . 4 ⊢ (tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
71	68, 69, 70	3syl 18	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
72		eqfnov2 7534	. . 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)))
73	64, 71, 72	syl2anc 583	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏)))
74	56, 73	mpbird 257	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  Vcvv 3468  ifcif 4523   × cxp 5667   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404   ∈ cmpo 7406  tpos ctpos 8208   ↑_m cmap 8819  Fincfn 8938  Basecbs 17150  0_gc0g 17391  1_rcur 20083  Ringcrg 20135  CRingccrg 20136   Mat cmat 22257   maDet cmdat 22436   maAdju cmadu 22484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-addf 11188  ax-mulf 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8144  df-tpos 8209  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-er 8702  df-map 8821  df-pm 8822  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-xnn0 12546  df-z 12560  df-dec 12679  df-uz 12824  df-rp 12978  df-fz 13488  df-fzo 13631  df-seq 13970  df-exp 14030  df-hash 14293  df-word 14468  df-lsw 14516  df-concat 14524  df-s1 14549  df-substr 14594  df-pfx 14624  df-splice 14703  df-reverse 14712  df-s2 14802  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-mulr 17217  df-starv 17218  df-sca 17219  df-vsca 17220  df-ip 17221  df-tset 17222  df-ple 17223  df-ds 17225  df-unif 17226  df-hom 17227  df-cco 17228  df-0g 17393  df-gsum 17394  df-prds 17399  df-pws 17401  df-mre 17536  df-mrc 17537  df-acs 17539  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-mhm 18710  df-submnd 18711  df-efmnd 18791  df-grp 18863  df-minusg 18864  df-mulg 18993  df-subg 19047  df-ghm 19136  df-gim 19181  df-cntz 19230  df-oppg 19259  df-symg 19284  df-pmtr 19359  df-psgn 19408  df-evpm 19409  df-cmn 19699  df-abl 19700  df-mgp 20037  df-rng 20055  df-ur 20084  df-ring 20137  df-cring 20138  df-oppr 20233  df-dvdsr 20256  df-unit 20257  df-invr 20287  df-dvr 20300  df-rhm 20371  df-subrng 20443  df-subrg 20468  df-drng 20586  df-sra 21018  df-rgmod 21019  df-cnfld 21236  df-zring 21329  df-zrh 21385  df-dsmm 21622  df-frlm 21637  df-mat 22258  df-mdet 22437  df-madu 22486

This theorem is referenced by:  madulid  22497