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Theorem submateq 29875

Description: Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)

**Hypotheses**
Ref	Expression
submateq.a	⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
submateq.b	⊢ 𝐵 = (Base‘𝐴)
submateq.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
submateq.i	⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
submateq.j	⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
submateq.e	⊢ (𝜑 → 𝐸 ∈ 𝐵)
submateq.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
submateq.1	⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))

**Assertion**
Ref	Expression
submateq	⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Distinct variable groups: 𝑖,𝐸,𝑗 𝑖,𝐹,𝑗 𝑖,𝐼,𝑗 𝑖,𝐽,𝑗 𝑖,𝑁,𝑗 𝜑,𝑖,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗) 𝐵(𝑖,𝑗) 𝑅(𝑖,𝑗)

Proof of Theorem submateq Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ (1...(𝑁 − 1)))
2		submateq.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑁 ∈ ℕ)
4		submateq.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
5	4	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝐼 ∈ (1...𝑁))
6		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑥 ∈ (1...(𝑁 − 1)))
7		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝐼 ≤ 𝑥)
8	3, 5, 6, 7	submateqlem1 29873	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → (𝑥 ∈ (𝐼...𝑁) ∧ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
9	8	simprd 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
10	1, 9	syldanl 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
11	10	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
12		simprr 796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ (1...(𝑁 − 1)))
13	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
14		submateq.j	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
15	14	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝐽 ∈ (1...𝑁))
16		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (1...(𝑁 − 1)))
17		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝐽 ≤ 𝑦)
18	13, 15, 16, 17	submateqlem1 29873	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → (𝑦 ∈ (𝐽...𝑁) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
19	18	simprd 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
20	12, 19	syldanl 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
21	20	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
22	11, 21	jca 554	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
23		ovexd 6680	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 + 1) ∈ V)
24		ovexd 6680	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 1) ∈ V)
25		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑖 = (𝑥 + 1))
26	25	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
27		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
28	27	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
29	26, 28	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))))
30		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸(𝑦 + 1)))
31		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹(𝑦 + 1)))
32	30, 31	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
33	29, 32	imbi12d 334	. . . . . . . . 9 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ↔ (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1)))))
34		submateq.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))
35	34	3expib 1268	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)))
36	23, 24, 33, 35	vtocl2d 29314	. . . . . . . 8 ⊢ (𝜑 → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
37	36	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
38	22, 37	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1)))
39		eqid 2622	. . . . . . 7 ⊢ (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐸)𝐽)
40	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
41	4	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝐼 ∈ (1...𝑁))
42	14	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝐽 ∈ (1...𝑁))
43		submateq.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
44		submateq.a	. . . . . . . . . 10 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
45		eqid 2622	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
46		submateq.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
47	44, 45, 46	matbas2i 20228	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝐵 → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
48	43, 47	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
49	48	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
50	8	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑥 ∈ (𝐼...𝑁))
51	1, 50	syldanl 735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) → 𝑥 ∈ (𝐼...𝑁))
52	51	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝑥 ∈ (𝐼...𝑁))
53	18	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
54	12, 53	syldanl 735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
55	54	adantlr 751	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
56	39, 40, 40, 41, 42, 49, 52, 55	smatbr 29867	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = ((𝑥 + 1)𝐸(𝑦 + 1)))
57		eqid 2622	. . . . . . 7 ⊢ (𝐼(subMat1‘𝐹)𝐽) = (𝐼(subMat1‘𝐹)𝐽)
58		submateq.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
59	44, 45, 46	matbas2i 20228	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
60	58, 59	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
61	60	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
62	57, 40, 40, 41, 42, 61, 52, 55	smatbr 29867	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = ((𝑥 + 1)𝐹(𝑦 + 1)))
63	38, 56, 62	3eqtr4d 2666	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
64	10	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
65	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
66	14	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
67		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1...(𝑁 − 1)))
68		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 < 𝐽)
69	65, 66, 67, 68	submateqlem2 29874	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → (𝑦 ∈ (1..^𝐽) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
70	69	simprd 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
71	12, 70	syldanl 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
72	71	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
73	64, 72	jca 554	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
74		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ V)
76		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑖 = (𝑥 + 1))
77	76	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
78		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑗 = 𝑦)
79		eqidd 2623	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((1...𝑁) ∖ {𝐽}) = ((1...𝑁) ∖ {𝐽}))
80	78, 79	eleq12d 2695	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
81	77, 80	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))))
82		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸𝑦))
83		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹𝑦))
84	82, 83	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
85	81, 84	imbi12d 334	. . . . . . . . 9 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ↔ (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦))))
86	23, 75, 85, 35	vtocl2d 29314	. . . . . . . 8 ⊢ (𝜑 → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
87	86	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
88	73, 87	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦))
89	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
90	4	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
91	14	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
92	48	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
93	51	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑥 ∈ (𝐼...𝑁))
94	69	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
95	12, 94	syldanl 735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
96	95	adantlr 751	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
97	39, 89, 89, 90, 91, 92, 93, 96	smattr 29865	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = ((𝑥 + 1)𝐸𝑦))
98	60	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
99	57, 89, 89, 90, 91, 98, 93, 96	smattr 29865	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = ((𝑥 + 1)𝐹𝑦))
100	88, 97, 99	3eqtr4d 2666	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
101		fz1ssnn 12372	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
102	101, 14	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ)
103	102	nnred 11035	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℝ)
104	103	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ ℝ)
105		fz1ssnn 12372	. . . . . . . . 9 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
106	105, 12	sseldi 3601	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℕ)
107	106	nnred 11035	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℝ)
108		lelttric 10144	. . . . . . 7 ⊢ ((𝐽 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐽 ≤ 𝑦 ∨ 𝑦 < 𝐽))
109	104, 107, 108	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝐽 ≤ 𝑦 ∨ 𝑦 < 𝐽))
110	109	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) → (𝐽 ≤ 𝑦 ∨ 𝑦 < 𝐽))
111	63, 100, 110	mpjaodan 827	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
112	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑁 ∈ ℕ)
113	4	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝐼 ∈ (1...𝑁))
114		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1...(𝑁 − 1)))
115		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 < 𝐼)
116	112, 113, 114, 115	submateqlem2 29874	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → (𝑥 ∈ (1..^𝐼) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
117	116	simprd 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
118	1, 117	syldanl 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
119	118	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
120	20	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
121	119, 120	jca 554	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
122		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
123	122	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ V)
124		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → 𝑖 = 𝑥)
125	124	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
126		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
127	126	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
128	125, 127	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))))
129		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = (𝑥𝐸(𝑦 + 1)))
130		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = (𝑥𝐹(𝑦 + 1)))
131	129, 130	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
132	128, 131	imbi12d 334	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ↔ ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1)))))
133	123, 24, 132, 35	vtocl2d 29314	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
134	133	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
135	121, 134	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1)))
136	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
137	4	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐼 ∈ (1...𝑁))
138	14	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐽 ∈ (1...𝑁))
139	48	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
140	116	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1..^𝐼))
141	1, 140	syldanl 735	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1..^𝐼))
142	141	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝑥 ∈ (1..^𝐼))
143	54	adantlr 751	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
144	39, 136, 136, 137, 138, 139, 142, 143	smatbl 29866	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥𝐸(𝑦 + 1)))
145	60	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
146	57, 136, 136, 137, 138, 145, 142, 143	smatbl 29866	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = (𝑥𝐹(𝑦 + 1)))
147	135, 144, 146	3eqtr4d 2666	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
148	118	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
149	71	adantlr 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
150	148, 149	jca 554	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
151		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → 𝑖 = 𝑥)
152	151	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
153		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → 𝑗 = 𝑦)
154	153	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
155	152, 154	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))))
156		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝐸𝑗) = (𝑥𝐸𝑦))
157		oveq12 6659	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝐹𝑗) = (𝑥𝐹𝑦))
158	156, 157	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
159	155, 158	imbi12d 334	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ↔ ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦))))
160	123, 75, 159, 35	vtocl2d 29314	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
161	160	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
162	150, 161	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦))
163	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
164	4	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
165	14	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
166	48	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐸 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
167	141	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑥 ∈ (1..^𝐼))
168	95	adantlr 751	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
169	39, 163, 163, 164, 165, 166, 167, 168	smattl 29864	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥𝐸𝑦))
170	60	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐹 ∈ ((Base‘𝑅) ↑_𝑚 ((1...𝑁) × (1...𝑁))))
171	57, 163, 163, 164, 165, 170, 167, 168	smattl 29864	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = (𝑥𝐹𝑦))
172	162, 169, 171	3eqtr4d 2666	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
173	109	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → (𝐽 ≤ 𝑦 ∨ 𝑦 < 𝐽))
174	147, 172, 173	mpjaodan 827	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
175	101, 4	sseldi 3601	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℕ)
176	175	nnred 11035	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
177	176	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ ℝ)
178	105, 1	sseldi 3601	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℕ)
179	178	nnred 11035	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℝ)
180		lelttric 10144	. . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐼 ≤ 𝑥 ∨ 𝑥 < 𝐼))
181	177, 179, 180	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝐼 ≤ 𝑥 ∨ 𝑥 < 𝐼))
182	111, 174, 181	mpjaodan 827	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
183	182	ralrimivva 2971	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...(𝑁 − 1))∀𝑦 ∈ (1...(𝑁 − 1))(𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
184		eqid 2622	. . . 4 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
185	44, 46, 184, 39, 2, 4, 14, 43	smatcl 29868	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
186	44, 46, 184, 57, 2, 4, 14, 58	smatcl 29868	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘𝐹)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
187		eqid 2622	. . . 4 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
188	187, 184	eqmat 20230	. . 3 ⊢ (((𝐼(subMat1‘𝐸)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) ∧ (𝐼(subMat1‘𝐹)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) → ((𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽) ↔ ∀𝑥 ∈ (1...(𝑁 − 1))∀𝑦 ∈ (1...(𝑁 − 1))(𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦)))
189	185, 186, 188	syl2anc 693	. 2 ⊢ (𝜑 → ((𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽) ↔ ∀𝑥 ∈ (1...(𝑁 − 1))∀𝑦 ∈ (1...(𝑁 − 1))(𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦)))
190	183, 189	mpbird 247	1 ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 {csn 4177 class class class wbr 4653 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ...cfz 12326 ..^cfzo 12465 Basecbs 15857 Mat cmat 20213 subMat1csmat 29859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-smat 29860

This theorem is referenced by: submatminr1 29876

W3C validator