Theorem smadiadetglem2 21385

Description: Lemma 2 for smadiadetg 21386. (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 5304	. . . . 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 21125	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6		elex 3428	. . . . . . 7 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ V)
7	6	adantr 484	. . . . . 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 19390	. . . . . 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 2758	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2758	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 19402	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17		eqid 2758	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	ring0cl 19403	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	ifcld 4469	. . . . 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 7269	. . . . 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 2759	. . . 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 7794	. . 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 19406	. . . . . . . . . 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 1132	. . . . . . . 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 4429	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
32	31	adantr 484	. . . . . . . 8 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
33	32	oveq2d 7172	. . . . . . 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 4429	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
35	34	adantr 484	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
36	30, 33, 35	3eqtr4d 2803	. . . . . 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 19422	. . . . . . . . . 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 1132	. . . . . . . 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 4432	. . . . . . . . 9 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
42	41	oveq2d 7172	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
43	42	adantr 484	. . . . . . 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 4432	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
45	44	adantr 484	. . . . . . 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 2803	. . . . . 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 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 7231	. . 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 2793	. 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 2758	. . . . . . 7 ⊢ (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
53	3, 4, 52, 15, 17	minmar1val 21361	. . . . . 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 5827	. . . 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 4701	. . . . . 6 ⊢ (𝐾 ∈ 𝑁 → {𝐾} ⊆ 𝑁)
57	56	3ad2ant2 1131	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58		ssid 3916	. . . . 5 ⊢ 𝑁 ⊆ 𝑁
59		resmpo 7272	. . . . 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 589	. . . 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 7268	. . . . 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 2797	. . 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 7172	. 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 2758	. . . . . 6 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
67	3, 4, 66, 17	marrepval 21275	. . . . 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 5827	. . 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 7272	. . . 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 589	. . 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 7268	. . . 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 2797	. 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 2804	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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3857   ⊆ wss 3860  ifcif 4423  {csn 4525   × cxp 5526   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158   ∘_f cof 7409  Fincfn 8540  Basecbs 16554  ._rcmulr 16637  0_gc0g 16784  1_rcur 19332  Ringcrg 19378  CRingccrg 19379   Mat cmat 21120   matRRep cmarrep 21269   maDet cmdat 21297   minMatR1 cminmar1 21346

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-plusg 16649  df-0g 16786  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-grp 18185  df-mgp 19321  df-ur 19333  df-ring 19380  df-cring 19381  df-mat 21121  df-marrep 21271  df-minmar1 21348

This theorem is referenced by:  smadiadetg  21386