Theorem smadiadetglem2 22590

Description: Lemma 2 for smadiadetg 22591. (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 5427	. . . . 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 22328	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6		elex 3482	. . . . . . 7 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ V)
7	6	adantr 479	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
8	5, 7	syl 17	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ V)
9	8	3ad2ant1 1130	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10		simp13 1202	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → 𝑆 ∈ (Base‘𝑅))
11		smadiadet.r	. . . . . 6 ⊢ 𝑅 ∈ CRing
12		crngring 20187	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	11, 12	mp1i 13	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
14		eqid 2725	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2725	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20204	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17		eqid 2725	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	ring0cl 20205	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	ifcld 4570	. . . . 5 ⊢ (𝑅 ∈ Ring → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
20	13, 19	syl 17	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
21		fconstmpo 7533	. . . . 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 2726	. . . 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 8089	. . 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 13	. . . . . . . . . 10 ⊢ (𝑆 ∈ (Base‘𝑅) → 𝑅 ∈ Ring)
26		smadiadetg.x	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
27	14, 26, 15	ringridm 20208	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
28	25, 27	mpancom 686	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r‘𝑅)) = 𝑆)
29	28	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
30	29	ad2antrl 726	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (1r‘𝑅)) = 𝑆)
31		iftrue 4530	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
32	31	adantr 479	. . . . . . . 8 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
33	32	oveq2d 7431	. . . . . . 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 4530	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
35	34	adantr 479	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
36	30, 33, 35	3eqtr4d 2775	. . . . . 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 20232	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
38	25, 37	mpancom 686	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
39	38	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
40	39	ad2antrl 726	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
41		iffalse 4533	. . . . . . . . 9 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
42	41	oveq2d 7431	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
43	42	adantr 479	. . . . . . 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 4533	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
45	44	adantr 479	. . . . . . 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 2775	. . . . . 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 810	. . . . 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 1128	. . . 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 7493	. . 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 2765	. 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 1134	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝐾 ∈ 𝑁)
52		eqid 2725	. . . . . . 7 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
53	3, 4, 52, 15, 17	minmar1val 22566	. . . . . 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 1406	. . . . 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 5978	. . . 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 4807	. . . . . 6 ⊢ (𝐾 ∈ 𝑁 → {𝐾} ⊆ 𝑁)
57	56	3ad2ant2 1131	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58		ssid 3995	. . . . 5 ⊢ 𝑁 ⊆ 𝑁
59		resmpo 7536	. . . . 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 584	. . . 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 7532	. . . . 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 2769	. . 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 7431	. 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 1146	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)))
66		eqid 2725	. . . . . 6 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
67	3, 4, 66, 17	marrepval 22480	. . . . 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 835	. . . 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 5978	. . 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 7536	. . . 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 584	. . 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 7532	. . . 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 2769	. 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 2776	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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∖ cdif 3937   ⊆ wss 3940  ifcif 4524  {csn 4624   × cxp 5670   ↾ cres 5674  ‘cfv 6542  (class class class)co 7415   ∈ cmpo 7417   ∘_f cof 7679  Fincfn 8960  Basecbs 17177  ._rcmulr 17231  0_gc0g 17418  1_rcur 20123  Ringcrg 20175  CRingccrg 20176   Mat cmat 22323   matRRep cmarrep 22474   maDet cmdat 22502   minMatR1 cminmar1 22551

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-plusg 17243  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-cmn 19739  df-abl 19740  df-mgp 20077  df-rng 20095  df-ur 20124  df-ring 20177  df-cring 20178  df-mat 22324  df-marrep 22476  df-minmar1 22553

This theorem is referenced by:  smadiadetg  22591