Theorem smadiadetglem2 22829

Description: Lemma 2 for smadiadetg 22830. (Contributed by AV, 14-Feb-2019.)

Hypotheses

Ref	Expression
smadiadet.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
smadiadet.b	⊢ 𝐵 = (Base‘𝐴)
smadiadet.r	⊢ 𝑅 ∈ CRing
smadiadet.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
smadiadet.h	⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)
smadiadetg.x	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
smadiadetglem2	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))

Proof of Theorem smadiadetglem2

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5410	. . . . 5 ⊢ {𝐾} ∈ V
2	1	a1i 11	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ∈ V)
3		smadiadet.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		smadiadet.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
5	3, 4	matrcl 22569	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6		elex 3476	. . . . . . 7 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ V)
7	6	adantr 485	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
8	5, 7	syl 18	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ V)
9	8	3ad2ant1 1151	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10		simp13 1224	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → 𝑆 ∈ (Base‘𝑅))
11		smadiadet.r	. . . . . 6 ⊢ 𝑅 ∈ CRing
12		crngring 20322	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	11, 12	mp1i 14	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
14		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2763	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20344	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17		eqid 2763	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	ring0cl 20346	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	ifcld 4534	. . . . 5 ⊢ (𝑅 ∈ Ring → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
20	13, 19	syl 18	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
21		fconstmpo 7527	. . . . 5 ⊢ (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ 𝑆)
22	21	a1i 11	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ 𝑆))
23		eqidd 2764	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))))
24	2, 9, 10, 20, 22, 23	offval22 8079	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))))
25	11, 12	mp1i 14	. . . . . . . . . 10 ⊢ (𝑆 ∈ (Base‘𝑅) → 𝑅 ∈ Ring)
26		smadiadetg.x	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
27	14, 26, 15	ringridm 20349	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
28	25, 27	mpancom 700	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r‘𝑅)) = 𝑆)
29	28	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
30	29	ad2antrl 740	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (1r‘𝑅)) = 𝑆)
31		iftrue 4493	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
32	31	adantr 485	. . . . . . . 8 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
33	32	oveq2d 7426	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (1r‘𝑅)))
34		iftrue 4493	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
35	34	adantr 485	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
36	30, 33, 35	3eqtr4d 2808	. . . . . 6 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)))
37	14, 26, 17	ringrz 20373	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
38	25, 37	mpancom 700	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
39	38	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
40	39	ad2antrl 740	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
41		iffalse 4496	. . . . . . . . 9 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
42	41	oveq2d 7426	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
43	42	adantr 485	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
44		iffalse 4496	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
45	44	adantr 485	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
46	40, 43, 45	3eqtr4d 2808	. . . . . 6 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)))
47	36, 46	pm2.61ian 823	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)))
48	47	3adant2 1149	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)))
49	48	mpoeq3dva 7487	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g‘𝑅))))
50	24, 49	eqtrd 2798	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g‘𝑅))))
51		simp2 1155	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝐾 ∈ 𝑁)
52		eqid 2763	. . . . . . 7 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
53	3, 4, 52, 15, 17	minmar1val 22805	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
54	51, 53	syld3an3 1436	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
55	54	reseq1d 5977	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)))
56		snssi 4751	. . . . . 6 ⊢ (𝐾 ∈ 𝑁 → {𝐾} ⊆ 𝑁)
57	56	3ad2ant2 1152	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58		ssid 3959	. . . . 5 ⊢ 𝑁 ⊆ 𝑁
59		resmpo 7530	. . . . 5 ⊢ (({𝐾} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
60	57, 58, 59	sylancl 597	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))))
61		mposnif 7526	. . . . 5 ⊢ (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))
62	61	a1i 11	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))))
63	55, 60, 62	3eqtrd 2802	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))))
64	63	oveq2d 7426	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)))))
65		3simpb 1167	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)))
66		eqid 2763	. . . . . 6 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
67	3, 4, 66, 17	marrepval 22719	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))))
68	65, 51, 51, 67	syl12anc 849	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))))
69	68	reseq1d 5977	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)))
70		resmpo 7530	. . . 4 ⊢ (({𝐾} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))))
71	57, 58, 70	sylancl 597	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))))
72		mposnif 7526	. . . 4 ⊢ (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)))
73	72	a1i 11	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g‘𝑅))))
74	69, 71, 73	3eqtrd 2802	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g‘𝑅))))
75	50, 64, 74	3eqtr4rd 2809	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ifcif 4487  {csn 4589   × cxp 5659   ↾ cres 5663  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ∘_f cof 7672  Fincfn 8939  Basecbs 17264  ._rcmulr 17306  0_gc0g 17487  1_rcur 20258  Ringcrg 20310  CRingccrg 20311   Mat cmat 22564   matRRep cmarrep 22713   maDet cmdat 22741   minMatR1 cminmar1 22790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-minusg 18999  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-mat 22565  df-marrep 22715  df-minmar1 22792

This theorem is referenced by:  smadiadetg  22830