Theorem smadiadetglem2 22699

Description: Lemma 2 for smadiadetg 22700. (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 5451	. . . . 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 22437	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6		elex 3509	. . . . . . 7 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ V)
7	6	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
8	5, 7	syl 17	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ V)
9	8	3ad2ant1 1133	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10		simp13 1205	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗 ∈ 𝑁) → 𝑆 ∈ (Base‘𝑅))
11		smadiadet.r	. . . . . 6 ⊢ 𝑅 ∈ CRing
12		crngring 20272	. . . . . 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 2740	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2740	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20289	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17		eqid 2740	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	ring0cl 20290	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	ifcld 4594	. . . . 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 7567	. . . . 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 2741	. . . 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 8129	. . 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 20293	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
28	25, 27	mpancom 687	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r‘𝑅)) = 𝑆)
29	28	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
30	29	ad2antrl 727	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (1r‘𝑅)) = 𝑆)
31		iftrue 4554	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
33	32	oveq2d 7464	. . . . . . 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 4554	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
36	30, 33, 35	3eqtr4d 2790	. . . . . 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 20317	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
38	25, 37	mpancom 687	. . . . . . . . 9 ⊢ (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
39	38	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
40	39	ad2antrl 727	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
41		iffalse 4557	. . . . . . . . 9 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
42	41	oveq2d 7464	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
43	42	adantr 480	. . . . . . 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 4557	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
45	44	adantr 480	. . . . . . 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 2790	. . . . . 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 811	. . . . 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 1131	. . . 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 7527	. . 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 2780	. 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 1137	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝐾 ∈ 𝑁)
52		eqid 2740	. . . . . . 7 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
53	3, 4, 52, 15, 17	minmar1val 22675	. . . . . 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 1409	. . . . 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 6008	. . . 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 4833	. . . . . 6 ⊢ (𝐾 ∈ 𝑁 → {𝐾} ⊆ 𝑁)
57	56	3ad2ant2 1134	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58		ssid 4031	. . . . 5 ⊢ 𝑁 ⊆ 𝑁
59		resmpo 7570	. . . . 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 585	. . . 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 7566	. . . . 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 2784	. . 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 7464	. 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 1149	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)))
66		eqid 2740	. . . . . 6 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
67	3, 4, 66, 17	marrepval 22589	. . . . 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 836	. . . 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 6008	. . 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 7570	. . . 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 585	. . 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 7566	. . . 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 2784	. 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 2791	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ifcif 4548  {csn 4648   × cxp 5698   ↾ cres 5702  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ∘_f cof 7712  Fincfn 9003  Basecbs 17258  ._rcmulr 17312  0_gc0g 17499  1_rcur 20208  Ringcrg 20260  CRingccrg 20261   Mat cmat 22432   matRRep cmarrep 22583   maDet cmdat 22611   minMatR1 cminmar1 22660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-mat 22433  df-marrep 22585  df-minmar1 22662

This theorem is referenced by:  smadiadetg  22700