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Theorem submateq 31084
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 (𝜑𝑁 ∈ ℕ)
submateq.i (𝜑𝐼 ∈ (1...𝑁))
submateq.j (𝜑𝐽 ∈ (1...𝑁))
submateq.e (𝜑𝐸𝐵)
submateq.f (𝜑𝐹𝐵)
submateq.1 ((𝜑𝑖 ∈ ((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.
StepHypRef Expression
1 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ (1...(𝑁 − 1)))
2 submateq.n . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
32ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → 𝑁 ∈ ℕ)
4 submateq.i . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (1...𝑁))
54ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → 𝐼 ∈ (1...𝑁))
6 simplr 768 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → 𝑥 ∈ (1...(𝑁 − 1)))
7 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → 𝐼𝑥)
83, 5, 6, 7submateqlem1 31082 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → (𝑥 ∈ (𝐼...𝑁) ∧ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
98simprd 499 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
101, 9syldanl 604 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
1110adantr 484 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
12 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ (1...(𝑁 − 1)))
132ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → 𝑁 ∈ ℕ)
14 submateq.j . . . . . . . . . . . . 13 (𝜑𝐽 ∈ (1...𝑁))
1514ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → 𝐽 ∈ (1...𝑁))
16 simplr 768 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → 𝑦 ∈ (1...(𝑁 − 1)))
17 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → 𝐽𝑦)
1813, 15, 16, 17submateqlem1 31082 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → (𝑦 ∈ (𝐽...𝑁) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
1918simprd 499 . . . . . . . . . 10 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
2012, 19syldanl 604 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
2120adantlr 714 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
2211, 21jca 515 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
23 ovexd 7165 . . . . . . . . 9 (𝜑 → (𝑥 + 1) ∈ V)
24 ovexd 7165 . . . . . . . . 9 (𝜑 → (𝑦 + 1) ∈ V)
25 simpl 486 . . . . . . . . . . . 12 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑖 = (𝑥 + 1))
2625eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
27 simpr 488 . . . . . . . . . . . 12 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
2827eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
2926, 28anbi12d 633 . . . . . . . . . 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 7139 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸(𝑦 + 1)))
31 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹(𝑦 + 1)))
3230, 31eqeq12d 2837 . . . . . . . . . 10 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
3329, 32imbi12d 348 . . . . . . . . 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 ((𝜑𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))
35343expib 1119 . . . . . . . . 9 (𝜑 → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)))
3623, 24, 33, 35vtocl2d 3534 . . . . . . . 8 (𝜑 → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
3736ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1))))
3822, 37mpd 15 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → ((𝑥 + 1)𝐸(𝑦 + 1)) = ((𝑥 + 1)𝐹(𝑦 + 1)))
39 eqid 2821 . . . . . . 7 (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐸)𝐽)
402ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝑁 ∈ ℕ)
414ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝐼 ∈ (1...𝑁))
4214ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝐽 ∈ (1...𝑁))
43 submateq.e . . . . . . . . 9 (𝜑𝐸𝐵)
44 submateq.a . . . . . . . . . 10 𝐴 = ((1...𝑁) Mat 𝑅)
45 eqid 2821 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
46 submateq.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
4744, 45, 46matbas2i 21006 . . . . . . . . 9 (𝐸𝐵𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
4843, 47syl 17 . . . . . . . 8 (𝜑𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
4948ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
508simpld 498 . . . . . . . . 9 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝐼𝑥) → 𝑥 ∈ (𝐼...𝑁))
511, 50syldanl 604 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) → 𝑥 ∈ (𝐼...𝑁))
5251adantr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝑥 ∈ (𝐼...𝑁))
5318simpld 498 . . . . . . . . 9 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝐽𝑦) → 𝑦 ∈ (𝐽...𝑁))
5412, 53syldanl 604 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐽𝑦) → 𝑦 ∈ (𝐽...𝑁))
5554adantlr 714 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝑦 ∈ (𝐽...𝑁))
5639, 40, 40, 41, 42, 49, 52, 55smatbr 31076 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = ((𝑥 + 1)𝐸(𝑦 + 1)))
57 eqid 2821 . . . . . . 7 (𝐼(subMat1‘𝐹)𝐽) = (𝐼(subMat1‘𝐹)𝐽)
58 submateq.f . . . . . . . . 9 (𝜑𝐹𝐵)
5944, 45, 46matbas2i 21006 . . . . . . . . 9 (𝐹𝐵𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
6058, 59syl 17 . . . . . . . 8 (𝜑𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
6160ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → 𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
6257, 40, 40, 41, 42, 61, 52, 55smatbr 31076 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = ((𝑥 + 1)𝐹(𝑦 + 1)))
6338, 56, 623eqtr4d 2866 . . . . 5 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
6410adantr 484 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}))
652ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
6614ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
67 simplr 768 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1...(𝑁 − 1)))
68 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 < 𝐽)
6965, 66, 67, 68submateqlem2 31083 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → (𝑦 ∈ (1..^𝐽) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
7069simprd 499 . . . . . . . . . 10 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
7112, 70syldanl 604 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
7271adantlr 714 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
7364, 72jca 515 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → ((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
74 vex 3474 . . . . . . . . . 10 𝑦 ∈ V
7574a1i 11 . . . . . . . . 9 (𝜑𝑦 ∈ V)
76 simpl 486 . . . . . . . . . . . 12 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑖 = (𝑥 + 1))
7776eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ (𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼})))
78 simpr 488 . . . . . . . . . . . 12 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → 𝑗 = 𝑦)
79 eqidd 2822 . . . . . . . . . . . 12 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((1...𝑁) ∖ {𝐽}) = ((1...𝑁) ∖ {𝐽}))
8078, 79eleq12d 2906 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
8177, 80anbi12d 633 . . . . . . . . . 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 7139 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐸𝑗) = ((𝑥 + 1)𝐸𝑦))
83 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → (𝑖𝐹𝑗) = ((𝑥 + 1)𝐹𝑦))
8482, 83eqeq12d 2837 . . . . . . . . . 10 ((𝑖 = (𝑥 + 1) ∧ 𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
8581, 84imbi12d 348 . . . . . . . . 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))))
8623, 75, 85, 35vtocl2d 3534 . . . . . . . 8 (𝜑 → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
8786ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → (((𝑥 + 1) ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦)))
8873, 87mpd 15 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → ((𝑥 + 1)𝐸𝑦) = ((𝑥 + 1)𝐹𝑦))
892ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
904ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
9114ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
9248ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
9351adantr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝑥 ∈ (𝐼...𝑁))
9469simpld 498 . . . . . . . . 9 (((𝜑𝑦 ∈ (1...(𝑁 − 1))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
9512, 94syldanl 604 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
9695adantlr 714 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
9739, 89, 89, 90, 91, 92, 93, 96smattr 31074 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = ((𝑥 + 1)𝐸𝑦))
9860ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → 𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
9957, 89, 89, 90, 91, 98, 93, 96smattr 31074 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = ((𝑥 + 1)𝐹𝑦))
10088, 97, 993eqtr4d 2866 . . . . 5 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
101 fz1ssnn 12921 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
102101, 14sseldi 3941 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ)
103102nnred 11630 . . . . . . . 8 (𝜑𝐽 ∈ ℝ)
104103adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ ℝ)
105 fz1ssnn 12921 . . . . . . . . 9 (1...(𝑁 − 1)) ⊆ ℕ
106105, 12sseldi 3941 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℕ)
107106nnred 11630 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑦 ∈ ℝ)
108 lelttric 10724 . . . . . . 7 ((𝐽 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐽𝑦𝑦 < 𝐽))
109104, 107, 108syl2anc 587 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝐽𝑦𝑦 < 𝐽))
110109adantr 484 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) → (𝐽𝑦𝑦 < 𝐽))
11163, 100, 110mpjaodan 956 . . . 4 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝐼𝑥) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
1122ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑁 ∈ ℕ)
1134ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝐼 ∈ (1...𝑁))
114 simplr 768 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1...(𝑁 − 1)))
115 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 < 𝐼)
116112, 113, 114, 115submateqlem2 31083 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → (𝑥 ∈ (1..^𝐼) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
117116simprd 499 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
1181, 117syldanl 604 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
119118adantr 484 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
12020adantlr 714 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))
121119, 120jca 515 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
122 vex 3474 . . . . . . . . . 10 𝑥 ∈ V
123122a1i 11 . . . . . . . . 9 (𝜑𝑥 ∈ V)
124 simpl 486 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → 𝑖 = 𝑥)
125124eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
126 simpr 488 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → 𝑗 = (𝑦 + 1))
127126eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})))
128125, 127anbi12d 633 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽}))))
129 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → (𝑖𝐸𝑗) = (𝑥𝐸(𝑦 + 1)))
130 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → (𝑖𝐹𝑗) = (𝑥𝐹(𝑦 + 1)))
131129, 130eqeq12d 2837 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = (𝑦 + 1)) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
132128, 131imbi12d 348 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = (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)))))
133123, 24, 132, 35vtocl2d 3534 . . . . . . . 8 (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
134133ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ (𝑦 + 1) ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1))))
135121, 134mpd 15 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑥𝐸(𝑦 + 1)) = (𝑥𝐹(𝑦 + 1)))
1362ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝑁 ∈ ℕ)
1374ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝐼 ∈ (1...𝑁))
13814ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝐽 ∈ (1...𝑁))
13948ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
140116simpld 498 . . . . . . . . 9 (((𝜑𝑥 ∈ (1...(𝑁 − 1))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1..^𝐼))
1411, 140syldanl 604 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1..^𝐼))
142141adantr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝑥 ∈ (1..^𝐼))
14354adantlr 714 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝑦 ∈ (𝐽...𝑁))
14439, 136, 136, 137, 138, 139, 142, 143smatbl 31075 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥𝐸(𝑦 + 1)))
14560ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → 𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
14657, 136, 136, 137, 138, 145, 142, 143smatbl 31075 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = (𝑥𝐹(𝑦 + 1)))
147135, 144, 1463eqtr4d 2866 . . . . 5 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝐽𝑦) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
148118adantr 484 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑥 ∈ ((1...𝑁) ∖ {𝐼}))
14971adantlr 714 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))
150148, 149jca 515 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
151 simpl 486 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = 𝑦) → 𝑖 = 𝑥)
152151eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ↔ 𝑥 ∈ ((1...𝑁) ∖ {𝐼})))
153 simpr 488 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = 𝑦) → 𝑗 = 𝑦)
154153eleq1d 2896 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑗 ∈ ((1...𝑁) ∖ {𝐽}) ↔ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})))
155152, 154anbi12d 633 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) ↔ (𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽}))))
156 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝐸𝑗) = (𝑥𝐸𝑦))
157 oveq12 7139 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝐹𝑗) = (𝑥𝐹𝑦))
158156, 157eqeq12d 2837 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑖𝐸𝑗) = (𝑖𝐹𝑗) ↔ (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
159155, 158imbi12d 348 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → (((𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ↔ ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦))))
160123, 75, 159, 35vtocl2d 3534 . . . . . . . 8 (𝜑 → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
161160ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → ((𝑥 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑦 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦)))
162150, 161mpd 15 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥𝐸𝑦) = (𝑥𝐹𝑦))
1632ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑁 ∈ ℕ)
1644ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐼 ∈ (1...𝑁))
16514ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐽 ∈ (1...𝑁))
16648ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐸 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
167141adantr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑥 ∈ (1..^𝐼))
16895adantlr 714 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝑦 ∈ (1..^𝐽))
16939, 163, 163, 164, 165, 166, 167, 168smattl 31073 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥𝐸𝑦))
17060ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → 𝐹 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
17157, 163, 163, 164, 165, 170, 167, 168smattl 31073 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) ∧ 𝑦 < 𝐽) → (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦) = (𝑥𝐹𝑦))
172162, 169, 1713eqtr4d 2866 . . . . 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))
173109adantr 484 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) ∧ 𝑥 < 𝐼) → (𝐽𝑦𝑦 < 𝐽))
174147, 172, 173mpjaodan 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))
175101, 4sseldi 3941 . . . . . . 7 (𝜑𝐼 ∈ ℕ)
176175nnred 11630 . . . . . 6 (𝜑𝐼 ∈ ℝ)
177176adantr 484 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ ℝ)
178105, 1sseldi 3941 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℕ)
179178nnred 11630 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → 𝑥 ∈ ℝ)
180 lelttric 10724 . . . . 5 ((𝐼 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐼𝑥𝑥 < 𝐼))
181177, 179, 180syl2anc 587 . . . 4 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝐼𝑥𝑥 < 𝐼))
182111, 174, 181mpjaodan 956 . . 3 ((𝜑 ∧ (𝑥 ∈ (1...(𝑁 − 1)) ∧ 𝑦 ∈ (1...(𝑁 − 1)))) → (𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
183182ralrimivva 3179 . 2 (𝜑 → ∀𝑥 ∈ (1...(𝑁 − 1))∀𝑦 ∈ (1...(𝑁 − 1))(𝑥(𝐼(subMat1‘𝐸)𝐽)𝑦) = (𝑥(𝐼(subMat1‘𝐹)𝐽)𝑦))
184 eqid 2821 . . . 4 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
18544, 46, 184, 39, 2, 4, 14, 43smatcl 31077 . . 3 (𝜑 → (𝐼(subMat1‘𝐸)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
18644, 46, 184, 57, 2, 4, 14, 58smatcl 31077 . . 3 (𝜑 → (𝐼(subMat1‘𝐹)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
187 eqid 2821 . . . 4 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
188187, 184eqmat 21008 . . 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)))
189185, 186, 188syl2anc 587 . 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)))
190183, 189mpbird 260 1 (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  cdif 3907  {csn 4540   class class class wbr 5039   × cxp 5526  cfv 6328  (class class class)co 7130  m cmap 8381  cr 10513  1c1 10515   + caddc 10517   < clt 10652  cle 10653  cmin 10847  cn 11615  ...cfz 12875  ..^cfzo 13016  Basecbs 16461   Mat cmat 20991  subMat1csmat 31068
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-ot 4549  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  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 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-sup 8882  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-fz 12876  df-fzo 13017  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-sca 16559  df-vsca 16560  df-ip 16561  df-tset 16562  df-ple 16563  df-ds 16565  df-hom 16567  df-cco 16568  df-0g 16693  df-prds 16699  df-pws 16701  df-sra 19919  df-rgmod 19920  df-dsmm 20851  df-frlm 20866  df-mat 20992  df-smat 31069
This theorem is referenced by:  submatminr1  31085
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