Theorem madutpos 21539

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 2737	. . . . . . . . 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 8005	. . . . . . . 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 870	. . . . . . . . . . 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 464	. . . . . . . . . . . 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 4462	. . . . . . . . . 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 7983	. . . . . . . . . . . 12 ⊢ (𝑐tpos 𝑀𝑑) = (𝑑𝑀𝑐)
9	8	eqcomi 2746	. . . . . . . . . . 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 4467	. . . . . . . . 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 7290	. . . . . . . 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	syl5eq 2790	. . . . . . 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 6721	. . . . . 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 767	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ CRing)
16		maduf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		maduf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
19	16, 18	matrcl 21309	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
20	19	simpld 498	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
21	20	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑁 ∈ Fin)
22		simp1ll 1238	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑅 ∈ CRing)
23		crngring 19574	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24		eqid 2737	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	17, 24	ringidcl 19586	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	17, 26	ring0cl 19587	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	25, 27	ifcld 4485	. . . . . . . . . 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 21319	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
31		elmapi 8530	. . . . . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
34	33	fovrnda 7379	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
35	34	3impb 1117	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
36	29, 35	ifcld 4485	. . . . . . . 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 21320	. . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅)
39	38, 16, 18	mdettpos 21508	. . . . . . 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 587	. . . . . 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 2779	. . . . 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 21350	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
43	42	adantl 485	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
44	43	adantr 484	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → tpos 𝑀 ∈ 𝐵)
45		simprl 771	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
46		simprr 773	. . . . . 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 21537	. . . . . 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 1378	. . . . 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 769	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
51	16, 38, 47, 18, 24, 26	maducoeval2 21537	. . . . . 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 1378	. . . . 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 2787	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎))
54		ovtpos 7983	. . . 4 ⊢ (𝑎tpos (𝐽‘𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎)
55	53, 54	eqtr4di 2796	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
56	55	ralrimivva 3112	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
57	16, 47, 18	maduf 21538	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
58	57	adantr 484	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:𝐵⟶𝐵)
59	58, 43	ffvelrnd 6905	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ 𝐵)
60	16, 17, 18	matbas2i 21319	. . . . 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 8530	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → (𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
63		ffn 6545	. . . 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	ffvelrnda 6904	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵)
66	16, 18	mattposcl 21350	. . . . 5 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ 𝐵)
67	16, 17, 18	matbas2i 21319	. . . . 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 8530	. . . 4 ⊢ (tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
70		ffn 6545	. . . 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 7340	. . 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 587	. 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 260	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408  ifcif 4439   × cxp 5549   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  tpos ctpos 7967   ↑_m cmap 8508  Fincfn 8626  Basecbs 16760  0_gc0g 16944  1_rcur 19516  Ringcrg 19562  CRingccrg 19563   Mat cmat 21304   maDet cmdat 21481   maAdju cmadu 21529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-addf 10808  ax-mulf 10809

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-tpos 7968  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-dec 12294  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-splice 14315  df-reverse 14324  df-s2 14413  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-0g 16946  df-gsum 16947  df-prds 16952  df-pws 16954  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-efmnd 18296  df-grp 18368  df-minusg 18369  df-mulg 18489  df-subg 18540  df-ghm 18620  df-gim 18663  df-cntz 18711  df-oppg 18738  df-symg 18760  df-pmtr 18834  df-psgn 18883  df-evpm 18884  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-rnghom 19735  df-drng 19769  df-subrg 19798  df-sra 20209  df-rgmod 20210  df-cnfld 20364  df-zring 20436  df-zrh 20470  df-dsmm 20694  df-frlm 20709  df-mat 21305  df-mdet 21482  df-madu 21531

This theorem is referenced by:  madulid  21542