Theorem mdetrlin2 22952

Description: The determinant function is additive for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)

Hypotheses

Ref	Expression
mdetrlin2.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetrlin2.k	⊢ 𝐾 = (Base‘𝑅)
mdetrlin2.p	⊢ + = (+g‘𝑅)
mdetrlin2.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetrlin2.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetrlin2.x	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
mdetrlin2.y	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
mdetrlin2.z	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
mdetrlin2.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)

Assertion

Ref	Expression
mdetrlin2	⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))))

Distinct variable groups:   𝜑,𝑖,𝑗   𝑖,𝐾,𝑗   𝑖,𝑁,𝑗   𝑖,𝐼,𝑗   + ,𝑖,𝑗

Allowed substitution hints:   𝐷(𝑖, 𝑗)   𝑅(𝑖, 𝑗)   𝑋(𝑖, 𝑗)   𝑌(𝑖, 𝑗)   𝑍(𝑖, 𝑗)

Proof of Theorem mdetrlin2

Step	Hyp	Ref	Expression
1		mdetrlin2.d	. 2 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		eqid 2763	. 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2763	. 2 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
4		mdetrlin2.p	. 2 ⊢ + = (+g‘𝑅)
5		mdetrlin2.r	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
6		mdetrlin2.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
7		mdetrlin2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		crngring 20502	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	5, 8	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	9	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
11		mdetrlin2.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
12		mdetrlin2.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
13	6, 4	ringacl 20537	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
14	10, 11, 12, 13	syl3anc 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑋 + 𝑌) ∈ 𝐾)
15		mdetrlin2.z	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
16	14, 15	ifcld 4532	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) ∈ 𝐾)
17	2, 6, 3, 7, 5, 16	matbas2d 22768	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ∈ (Base‘(𝑁 Mat 𝑅)))
18	11, 15	ifcld 4532	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, 𝑍) ∈ 𝐾)
19	2, 6, 3, 7, 5, 18	matbas2d 22768	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ∈ (Base‘(𝑁 Mat 𝑅)))
20	12, 15	ifcld 4532	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑌, 𝑍) ∈ 𝐾)
21	2, 6, 3, 7, 5, 20	matbas2d 22768	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ∈ (Base‘(𝑁 Mat 𝑅)))
22		mdetrlin2.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
23		snex 5401	. . . . . . 7 ⊢ {𝐼} ∈ V
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝐼} ∈ V)
25	22	snssd 4747	. . . . . . . . 9 ⊢ (𝜑 → {𝐼} ⊆ 𝑁)
26	25	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝐼} ∧ 𝑗 ∈ 𝑁) → {𝐼} ⊆ 𝑁)
27		simp2 1155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝐼} ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ {𝐼})
28	26, 27	sseldd 3914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝐼} ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
29	28, 11	syld3an2 1439	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝐼} ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
30	28, 12	syld3an2 1439	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝐼} ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
31		eqidd 2764	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋))
32		eqidd 2764	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌))
33	24, 7, 29, 30, 31, 32	offval22 8124	. . . . 5 ⊢ (𝜑 → ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ (𝑋 + 𝑌)))
34	33	eqcomd 2769	. . . 4 ⊢ (𝜑 → (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ (𝑋 + 𝑌)) = ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌)))
35		mposnif 7559	. . . 4 ⊢ (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ (𝑋 + 𝑌))
36		mposnif 7559	. . . . 5 ⊢ (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋)
37		mposnif 7559	. . . . 5 ⊢ (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌)
38	36, 37	oveq12i 7455	. . . 4 ⊢ ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍))) = ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑋) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ 𝑌))
39	34, 35, 38	3eqtr4g 2823	. . 3 ⊢ (𝜑 → (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) = ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍))))
40		ssid 3935	. . . 4 ⊢ 𝑁 ⊆ 𝑁
41		resmpo 7563	. . . 4 ⊢ (({𝐼} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)))
42	25, 40, 41	sylancl 598	. . 3 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)))
43		resmpo 7563	. . . . 5 ⊢ (({𝐼} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)))
44	25, 40, 43	sylancl 598	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)))
45		resmpo 7563	. . . . 5 ⊢ (({𝐼} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))
46	25, 40, 45	sylancl 598	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ({𝐼} × 𝑁)) = (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))
47	44, 46	oveq12d 7461	. . 3 ⊢ (𝜑 → (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ({𝐼} × 𝑁)) ∘f + ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ({𝐼} × 𝑁))) = ((𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ∘f + (𝑖 ∈ {𝐼}, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍))))
48	39, 42, 47	3eqtr4d 2808	. 2 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ({𝐼} × 𝑁)) = (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ({𝐼} × 𝑁)) ∘f + ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ({𝐼} × 𝑁))))
49		eldifsni 4753	. . . . . . 7 ⊢ (𝑖 ∈ (𝑁 ∖ {𝐼}) → 𝑖 ≠ 𝐼)
50	49	neneqd 2964	. . . . . 6 ⊢ (𝑖 ∈ (𝑁 ∖ {𝐼}) → ¬ 𝑖 = 𝐼)
51		iffalse 4494	. . . . . . 7 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = 𝑍)
52		iffalse 4494	. . . . . . 7 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑋, 𝑍) = 𝑍)
53	51, 52	eqtr4d 2801	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑋, 𝑍))
54	50, 53	syl 18	. . . . 5 ⊢ (𝑖 ∈ (𝑁 ∖ {𝐼}) → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑋, 𝑍))
55	54	3ad2ant2 1152	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑋, 𝑍))
56	55	mpoeq3dva 7520	. . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)))
57		difss 4075	. . . 4 ⊢ (𝑁 ∖ {𝐼}) ⊆ 𝑁
58		resmpo 7563	. . . 4 ⊢ (((𝑁 ∖ {𝐼}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)))
59	57, 40, 58	mp2an 705	. . 3 ⊢ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))
60		resmpo 7563	. . . 4 ⊢ (((𝑁 ∖ {𝐼}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)))
61	57, 40, 60	mp2an 705	. . 3 ⊢ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))
62	56, 59, 61	3eqtr4g 2823	. 2 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
63		iffalse 4494	. . . . . . 7 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑌, 𝑍) = 𝑍)
64	51, 63	eqtr4d 2801	. . . . . 6 ⊢ (¬ 𝑖 = 𝐼 → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑌, 𝑍))
65	50, 64	syl 18	. . . . 5 ⊢ (𝑖 ∈ (𝑁 ∖ {𝐼}) → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑌, 𝑍))
66	65	3ad2ant2 1152	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍) = if(𝑖 = 𝐼, 𝑌, 𝑍))
67	66	mpoeq3dva 7520	. . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))
68		resmpo 7563	. . . 4 ⊢ (((𝑁 ∖ {𝐼}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))
69	57, 40, 68	mp2an 705	. . 3 ⊢ ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍))
70	67, 59, 69	3eqtr4g 2823	. 2 ⊢ (𝜑 → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)) ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))
71	1, 2, 3, 4, 5, 17, 19, 21, 22, 48, 62, 70	mdetrlin 22947	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ifcif 4485  {csn 4588   × cxp 5653   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   ∘_f cof 7715  Fincfn 9001  Basecbs 17416  +_gcplusg 17457  Ringcrg 20489  CRingccrg 20490   Mat cmat 22752   maDet cmdat 22929

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-addf 11306  ax-mulf 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  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-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-tpos 8271  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-xnn0 12707  df-z 12721  df-dec 12842  df-uz 12993  df-rp 13148  df-fz 13667  df-fzo 13816  df-seq 14172  df-exp 14232  df-hash 14502  df-word 14686  df-lsw 14735  df-concat 14743  df-s1 14770  df-substr 14817  df-pfx 14849  df-splice 14927  df-reverse 14936  df-s2 15027  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-starv 17472  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-unif 17480  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-efmnd 19094  df-grp 19176  df-minusg 19177  df-mulg 19307  df-subg 19362  df-ghm 19457  df-gim 19502  df-cntz 19561  df-oppg 19590  df-symg 19614  df-pmtr 19686  df-psgn 19735  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-cring 20492  df-oppr 20597  df-dvdsr 20617  df-unit 20618  df-invr 20648  df-dvr 20661  df-rhm 20732  df-subrng 20828  df-subrg 20852  df-drng 21012  df-sra 21478  df-rgmod 21479  df-cnfld 21709  df-zring 21783  df-zrh 21839  df-dsmm 22068  df-frlm 22083  df-mat 22753  df-mdet 22930

This theorem is used by:  mdetero  22955  madugsum  22988