Theorem madutpos 22625

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 2739	. . . . . . . . 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 8203	. . . . . . . 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 876	. . . . . . . . . . 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 461	. . . . . . . . . . . 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 4478	. . . . . . . . . 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 8181	. . . . . . . . . . . 12 ⊢ (𝑐tpos 𝑀𝑑) = (𝑑𝑀𝑐)
9	8	eqcomi 2748	. . . . . . . . . . 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 4483	. . . . . . . . 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 7435	. . . . . . . 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 2786	. . . . . . 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 6831	. . . . . 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 772	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ CRing)
16		maduf.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		maduf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
19	16, 18	matrcl 22395	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
20	19	simpld 495	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
21	20	ad2antlr 733	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑁 ∈ Fin)
22		simp1ll 1243	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑅 ∈ CRing)
23		crngring 20217	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24		eqid 2739	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	17, 24	ringidcl 20237	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26		eqid 2739	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
27	17, 26	ring0cl 20239	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	25, 27	ifcld 4501	. . . . . . . . . 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 22405	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
31		elmapi 8786	. . . . . . . . . . . . 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 733	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
34	33	fovcdmda 7527	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
35	34	3impb 1120	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
36	29, 35	ifcld 4501	. . . . . . . 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 22406	. . . . . . 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 2739	. . . . . . . 8 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅)
39	38, 16, 18	mdettpos 22594	. . . . . . 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 590	. . . . . 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 2776	. . . . 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 22436	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
43	42	adantl 482	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
44	43	adantr 481	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → tpos 𝑀 ∈ 𝐵)
45		simprl 776	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
46		simprr 778	. . . . . 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 22623	. . . . . 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 1384	. . . . 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 774	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
51	16, 38, 47, 18, 24, 26	maducoeval2 22623	. . . . . 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 1384	. . . . 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 2784	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎))
54		ovtpos 8181	. . . 4 ⊢ (𝑎tpos (𝐽‘𝑀)𝑏) = (𝑏(𝐽‘𝑀)𝑎)
55	53, 54	eqtr4di 2792	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
56	55	ralrimivva 3182	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝐽‘tpos 𝑀)𝑏) = (𝑎tpos (𝐽‘𝑀)𝑏))
57	16, 47, 18	maduf 22624	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
58	57	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:𝐵⟶𝐵)
59	58, 43	ffvelcdmd 7026	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) ∈ 𝐵)
60	16, 17, 18	matbas2i 22405	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ 𝐵 → (𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
61		elmapi 8786	. . . 4 ⊢ ((𝐽‘tpos 𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → (𝐽‘tpos 𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
62		ffn 6655	. . . 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 7025	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵)
65	16, 18	mattposcl 22436	. . . . 5 ⊢ ((𝐽‘𝑀) ∈ 𝐵 → tpos (𝐽‘𝑀) ∈ 𝐵)
66	16, 17, 18	matbas2i 22405	. . . . 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 8786	. . . 4 ⊢ (tpos (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → tpos (𝐽‘𝑀):(𝑁 × 𝑁)⟶(Base‘𝑅))
69		ffn 6655	. . . 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 7486	. . 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 590	. 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 258	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431  ifcif 4454   × cxp 5616   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  tpos ctpos 8165   ↑_m cmap 8763  Fincfn 8883  Basecbs 17170  0_gc0g 17393  1_rcur 20153  Ringcrg 20205  CRingccrg 20206   Mat cmat 22390   maDet cmdat 22567   maAdju cmadu 22615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-addf 11108  ax-mulf 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-xor 1519  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-xnn0 12502  df-z 12516  df-dec 12636  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-hash 14284  df-word 14467  df-lsw 14516  df-concat 14524  df-s1 14550  df-substr 14595  df-pfx 14625  df-splice 14703  df-reverse 14712  df-s2 14801  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-efmnd 18828  df-grp 18903  df-minusg 18904  df-mulg 19035  df-subg 19090  df-ghm 19179  df-gim 19225  df-cntz 19283  df-oppg 19312  df-symg 19336  df-pmtr 19408  df-psgn 19457  df-evpm 19458  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-dvr 20372  df-rhm 20443  df-subrng 20518  df-subrg 20542  df-drng 20703  df-sra 21163  df-rgmod 21164  df-cnfld 21348  df-zring 21422  df-zrh 21478  df-dsmm 21707  df-frlm 21722  df-mat 22391  df-mdet 22568  df-madu 22617

This theorem is referenced by:  madulid  22628