Theorem madutpos 22620

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 8207	. . . . . . . 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 871	. . . . . . . . . . 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 4491	. . . . . . . . . 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 8185	. . . . . . . . . . . 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 4496	. . . . . . . . 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 7440	. . . . . . . 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 2784	. . . . . . 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 6839	. . . . . 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 22390	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
20	19	simpld 494	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
21	20	ad2antlr 728	. . . . . . . 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 20220	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24		eqid 2737	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	17, 24	ringidcl 20240	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	17, 26	ring0cl 20242	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	25, 27	ifcld 4514	. . . . . . . . . 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 22400	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
31		elmapi 8790	. . . . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
34	33	fovcdmda 7532	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
35	34	3impb 1115	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
36	29, 35	ifcld 4514	. . . . . . . 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 22401	. . . . . . 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 22589	. . . . . . 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 585	. . . . . 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 2774	. . . . 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 22431	. . . . . . . 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 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 22618	. . . . . 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 1379	. . . . 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 22618	. . . . . 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 1379	. . . . 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 2782	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎))
54		ovtpos 8185	. . . 4 ⊢ (𝑎tpos (𝐽‘𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎)
55	53, 54	eqtr4di 2790	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
56	55	ralrimivva 3181	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
57	16, 47, 18	maduf 22619	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
58	57	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:𝐵⟶𝐵)
59	58, 43	ffvelcdmd 7032	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ 𝐵)
60	16, 17, 18	matbas2i 22400	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ 𝐵 → (𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
61		elmapi 8790	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → (𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
62		ffn 6663	. . . 4 ⊢ ((𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
63	59, 60, 61, 62	4syl 19	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) Fn (𝑁 × 𝑁))
64	57	ffvelcdmda 7031	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵)
65	16, 18	mattposcl 22431	. . . . 5 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ 𝐵)
66	16, 17, 18	matbas2i 22400	. . . . 5 ⊢ (tpos (𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
67	64, 65, 66	3syl 18	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
68		elmapi 8790	. . . 4 ⊢ (tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
69		ffn 6663	. . . 4 ⊢ (tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
70	67, 68, 69	3syl 18	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos (𝐽‘𝑀) Fn (𝑁 × 𝑁))
71		eqfnov2 7491	. . 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 585	. 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 257	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ifcif 4467   × cxp 5623   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  tpos ctpos 8169   ↑_m cmap 8767  Fincfn 8887  Basecbs 17173  0_gc0g 17396  1_rcur 20156  Ringcrg 20208  CRingccrg 20209   Mat cmat 22385   maDet cmdat 22562   maAdju cmadu 22610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-addf 11111  ax-mulf 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-xnn0 12505  df-z 12519  df-dec 12639  df-uz 12783  df-rp 12937  df-fz 13456  df-fzo 13603  df-seq 13958  df-exp 14018  df-hash 14287  df-word 14470  df-lsw 14519  df-concat 14527  df-s1 14553  df-substr 14598  df-pfx 14628  df-splice 14706  df-reverse 14715  df-s2 14804  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-efmnd 18831  df-grp 18906  df-minusg 18907  df-mulg 19038  df-subg 19093  df-ghm 19182  df-gim 19228  df-cntz 19286  df-oppg 19315  df-symg 19339  df-pmtr 19411  df-psgn 19460  df-evpm 19461  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-oppr 20311  df-dvdsr 20331  df-unit 20332  df-invr 20362  df-dvr 20375  df-rhm 20446  df-subrng 20517  df-subrg 20541  df-drng 20702  df-sra 21163  df-rgmod 21164  df-cnfld 21348  df-zring 21440  df-zrh 21496  df-dsmm 21725  df-frlm 21740  df-mat 22386  df-mdet 22563  df-madu 22612

This theorem is referenced by:  madulid  22623