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Theorem submateq 31759

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 768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ (1...(𝑁 − 1)))
2		submateq.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑁 ∈ ℕ)
4		submateq.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
5	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝐼 ∈ (1...𝑁))
6		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑥 ∈ (1...(𝑁 − 1)))
7		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝐼 ≤ 𝑥)
8	3, 5, 6, 7	submateqlem1 31757	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → (𝑥 ∈ (𝐼...𝑁) ∧ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
9	8	simprd 496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
10	1, 9	syldanl 602	. . . . . . . . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ (1...(𝑁 − 1)))
13	2	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
14		submateq.j	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
15	14	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝐽 ∈ (1...𝑁))
16		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (1...(𝑁 − 1)))
17		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝐽 ≤ 𝑦)
18	13, 15, 16, 17	submateqlem1 31757	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → (𝑦 ∈ (𝐽...𝑁) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
19	18	simprd 496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
20	12, 19	syldanl 602	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
21	20	adantlr 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
22	11, 21	jca 512	. . . . . . 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 7310	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 + 1) ∈ V)
24		ovexd 7310	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 1) ∈ V)
25		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑖 = (𝑥 + 1))
26	25	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
27		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
28	27	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
29	26, 28	anbi12d 631	. . . . . . . . . 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 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸(𝑦 + 1)))
31		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹(𝑦 + 1)))
32	30, 31	eqeq12d 2754	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
33	29, 32	imbi12d 345	. . . . . . . . 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 1121	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)))
36	23, 24, 33, 35	vtocl2d 3496	. . . . . . . 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 727	. . . . . . 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 2738	. . . . . . 7 ⊢ (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐸)𝐽)
40	2	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
41	4	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝐽 ≤ 𝑦) → 𝐼 ∈ (1...𝑁))
42	14	ad3antrrr 727	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
46		submateq.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
47	44, 45, 46	matbas2i 21571	. . . . . . . . 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 727	. . . . . . 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 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼 ≤ 𝑥) → 𝑥 ∈ (𝐼...𝑁))
51	1, 50	syldanl 602	. . . . . . . 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 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
54	12, 53	syldanl 602	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽 ≤ 𝑦) → 𝑦 ∈ (𝐽...𝑁))
55	54	adantlr 712	. . . . . . 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 31751	. . . . . 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 2738	. . . . . . 7 ⊢ (𝐼(subMat1‘𝐹)𝐽) = (𝐼(subMat1‘𝐹)𝐽)
58		submateq.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
59	44, 45, 46	matbas2i 21571	. . . . . . . . 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 727	. . . . . . 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 31751	. . . . . 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 2788	. . . . 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 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
66	14	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
67		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1...(𝑁 − 1)))
68		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 < 𝐽)
69	65, 66, 67, 68	submateqlem2 31758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → (𝑦 ∈ (1..^𝐽) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
70	69	simprd 496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
71	12, 70	syldanl 602	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
72	71	adantlr 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
73	64, 72	jca 512	. . . . . . 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 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ V)
76		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑖 = (𝑥 + 1))
77	76	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
78		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑗 = 𝑦)
79		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((1...𝑁) ∖ {𝐽}) = ((1...𝑁) ∖ {𝐽}))
80	78, 79	eleq12d 2833	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
81	77, 80	anbi12d 631	. . . . . . . . . 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 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸𝑦))
83		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹𝑦))
84	82, 83	eqeq12d 2754	. . . . . . . . . 10 ⊢ ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
85	81, 84	imbi12d 345	. . . . . . . . 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 3496	. . . . . . . 8 ⊢ (𝜑 → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
87	86	ad3antrrr 727	. . . . . . 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 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
90	4	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
91	14	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼 ≤ 𝑥) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
92	48	ad3antrrr 727	. . . . . . 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 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
95	12, 94	syldanl 602	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
96	95	adantlr 712	. . . . . . 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 31749	. . . . . 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 727	. . . . . . 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 31749	. . . . . 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 2788	. . . . 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 13287	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
102	101, 14	sselid 3919	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ)
103	102	nnred 11988	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℝ)
104	103	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ ℝ)
105		fz1ssnn 13287	. . . . . . . . 9 ⊢ (1...(𝑁 − 1)) ⊆ ℕ
106	105, 12	sselid 3919	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℕ)
107	106	nnred 11988	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℝ)
108		lelttric 11082	. . . . . . 7 ⊢ ((𝐽 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐽 ≤ 𝑦 ∨ 𝑦 < 𝐽))
109	104, 107, 108	syl2anc 584	. . . . . 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 956	. . . 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 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑁 ∈ ℕ)
113	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝐼 ∈ (1...𝑁))
114		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1...(𝑁 − 1)))
115		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 < 𝐼)
116	112, 113, 114, 115	submateqlem2 31758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → (𝑥 ∈ (1..^𝐼) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
117	116	simprd 496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
118	1, 117	syldanl 602	. . . . . . . . 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 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
121	119, 120	jca 512	. . . . . . 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 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
123	122	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ V)
124		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → 𝑖 = 𝑥)
125	124	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
126		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
127	126	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
128	125, 127	anbi12d 631	. . . . . . . . . 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 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = (𝑥𝐸(𝑦 + 1)))
130		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = (𝑥𝐹(𝑦 + 1)))
131	129, 130	eqeq12d 2754	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
132	128, 131	imbi12d 345	. . . . . . . . 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 3496	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
134	133	ad3antrrr 727	. . . . . . 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 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝑁 ∈ ℕ)
137	4	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐼 ∈ (1...𝑁))
138	14	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽 ≤ 𝑦) → 𝐽 ∈ (1...𝑁))
139	48	ad3antrrr 727	. . . . . . 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 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1..^𝐼))
141	1, 140	syldanl 602	. . . . . . . 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 712	. . . . . . 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 31750	. . . . . 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 727	. . . . . . 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 31750	. . . . . 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 2788	. . . . 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 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
150	148, 149	jca 512	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
151		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → 𝑖 = 𝑥)
152	151	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
153		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → 𝑗 = 𝑦)
154	153	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
155	152, 154	anbi12d 631	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))))
156		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝐸𝑗) = (𝑥𝐸𝑦))
157		oveq12 7284	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝐹𝑗) = (𝑥𝐹𝑦))
158	156, 157	eqeq12d 2754	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
159	155, 158	imbi12d 345	. . . . . . . . 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 3496	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
161	160	ad3antrrr 727	. . . . . . 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 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
164	4	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
165	14	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
166	48	ad3antrrr 727	. . . . . . 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 712	. . . . . . 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 31748	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥𝐸𝑦))
170	60	ad3antrrr 727	. . . . . . 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 31748	. . . . . 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 2788	. . . . 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 956	. . . 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	sselid 3919	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℕ)
176	175	nnred 11988	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
177	176	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ ℝ)
178	105, 1	sselid 3919	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℕ)
179	178	nnred 11988	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℝ)
180		lelttric 11082	. . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐼 ≤ 𝑥 ∨ 𝑥 < 𝐼))
181	177, 179, 180	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝐼 ≤ 𝑥 ∨ 𝑥 < 𝐼))
182	111, 174, 181	mpjaodan 956	. . 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 3123	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...(𝑁 − 1))∀𝑦 ∈ (1...(𝑁 − 1))(𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
184		eqid 2738	. . . 4 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
185	44, 46, 184, 39, 2, 4, 14, 43	smatcl 31752	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
186	44, 46, 184, 57, 2, 4, 14, 58	smatcl 31752	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘𝐹)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
187		eqid 2738	. . . 4 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
188	187, 184	eqmat 21573	. . 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 584	. 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 256	1 ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ∖ cdif 3884 {csn 4561 class class class wbr 5074 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℝcr 10870 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ...cfz 13239 ..^cfzo 13382 Basecbs 16912 Mat cmat 21554 subMat1csmat 31743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-pws 17160 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-mat 21555 df-smat 31744

This theorem is referenced by: submatminr1 31760

W3C validator