Theorem smadiadetglem2 23031

Description: Lemma 2 for smadiadetg 23032. (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 5402	. . . . 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 22771	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6		elex 3475	. . . . . . 7 ⊢ (𝑁 ∈ Fin → 𝑁 ∈ V)
7	6	adantr 486	. . . . . 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 20514	. . . . . 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 20536	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17		eqid 2763	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	ring0cl 20538	. . . . . 6 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	ifcld 4532	. . . . 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 7566	. . . . 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 8132	. . 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 20541	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r‘𝑅)) = 𝑆)
28	25, 27	mpancom 701	. . . . . . . . 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 741	. . . . . . 7 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (1r‘𝑅)) = 𝑆)
31		iftrue 4491	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
32	31	adantr 486	. . . . . . . 8 ⊢ ((𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅))
33	32	oveq2d 7465	. . . . . . 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 4491	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = 𝑆)
35	34	adantr 486	. . . . . . 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 20567	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
38	25, 37	mpancom 701	. . . . . . . . 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 741	. . . . . . 7 ⊢ ((¬ 𝑗 = 𝐾 ∧ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ 𝑗 ∈ 𝑁)) → (𝑆 · (0g‘𝑅)) = (0g‘𝑅))
41		iffalse 4494	. . . . . . . . 9 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
42	41	oveq2d 7465	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r‘𝑅), (0g‘𝑅))) = (𝑆 · (0g‘𝑅)))
43	42	adantr 486	. . . . . . 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 4494	. . . . . . . 8 ⊢ (¬ 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g‘𝑅)) = (0g‘𝑅))
45	44	adantr 486	. . . . . . 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 824	. . . . 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 7526	. . 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 23007	. . . . . 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 1437	. . . . 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 5983	. . . 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 4746	. . . . . 6 ⊢ (𝐾 ∈ 𝑁 → {𝐾} ⊆ 𝑁)
57	56	3ad2ant2 1152	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58		ssid 3935	. . . . 5 ⊢ 𝑁 ⊆ 𝑁
59		resmpo 7569	. . . . 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 598	. . . 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 7565	. . . . 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 7465	. 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 22921	. . . . 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 850	. . . 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 5983	. . 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 7569	. . . 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 598	. . 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 7565	. . . 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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ifcif 4485  {csn 4588   × cxp 5654   ↾ cres 5658  ‘cfv 6564  (class class class)co 7449   ∈ cmpo 7451   ∘_f cof 7721  Fincfn 9009  Basecbs 17428  ._rcmulr 17470  0_gc0g 17651  1_rcur 20449  Ringcrg 20501  CRingccrg 20502   Mat cmat 22766   matRRep cmarrep 22915   maDet cmdat 22943   minMatR1 cminmar1 22992

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-2 12443  df-sets 17383  df-slot 17401  df-ndx 17413  df-base 17429  df-plusg 17482  df-0g 17653  df-mgm 18857  df-sgrp 18949  df-mnd 18965  df-grp 19188  df-minusg 19189  df-cmn 20038  df-abl 20039  df-mgp 20403  df-rng 20417  df-ur 20450  df-ring 20503  df-cring 20504  df-mat 22767  df-marrep 22917  df-minmar1 22994

This theorem is used by:  smadiadetg  23032