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Theorem smatrcl 29024
Description: Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
Hypotheses
Ref Expression
smat.s 𝑆 = (𝐾(subMat1‘𝐴)𝐿)
smat.m (𝜑𝑀 ∈ ℕ)
smat.n (𝜑𝑁 ∈ ℕ)
smat.k (𝜑𝐾 ∈ (1...𝑀))
smat.l (𝜑𝐿 ∈ (1...𝑁))
smat.a (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))
Assertion
Ref Expression
smatrcl (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Proof of Theorem smatrcl
Dummy variables 𝑖 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smat.a . . . . . . . 8 (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))
2 elmapi 7743 . . . . . . . 8 (𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3 ffun 5947 . . . . . . . 8 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (𝜑 → Fun 𝐴)
5 eqid 2609 . . . . . . . . 9 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
65mpt2fun 6638 . . . . . . . 8 Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
76a1i 11 . . . . . . 7 (𝜑 → Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
8 funco 5828 . . . . . . 7 ((Fun 𝐴 ∧ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) → Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
94, 7, 8syl2anc 690 . . . . . 6 (𝜑 → Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
10 smat.s . . . . . . . 8 𝑆 = (𝐾(subMat1‘𝐴)𝐿)
11 fz1ssnn 12201 . . . . . . . . . 10 (1...𝑀) ⊆ ℕ
12 smat.k . . . . . . . . . 10 (𝜑𝐾 ∈ (1...𝑀))
1311, 12sseldi 3565 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ)
14 fz1ssnn 12201 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
15 smat.l . . . . . . . . . 10 (𝜑𝐿 ∈ (1...𝑁))
1614, 15sseldi 3565 . . . . . . . . 9 (𝜑𝐿 ∈ ℕ)
17 smatfval 29023 . . . . . . . . 9 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁)))) → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1813, 16, 1, 17syl3anc 1317 . . . . . . . 8 (𝜑 → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1910, 18syl5eq 2655 . . . . . . 7 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 5811 . . . . . 6 (𝜑 → (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
219, 20mpbird 245 . . . . 5 (𝜑 → Fun 𝑆)
22 fdmrn 5963 . . . . 5 (Fun 𝑆𝑆:dom 𝑆⟶ran 𝑆)
2321, 22sylib 206 . . . 4 (𝜑𝑆:dom 𝑆⟶ran 𝑆)
2419dmeqd 5235 . . . . . 6 (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 5546 . . . . . . 7 dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ dom 𝐴)
26 fdm 5950 . . . . . . . . . . . 12 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
271, 2, 263syl 18 . . . . . . . . . . 11 (𝜑 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
2827imaeq2d 5372 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ dom 𝐴) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ ((1...𝑀) × (1...𝑁))))
2928eleq2d 2672 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ dom 𝐴) ↔ 𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ ((1...𝑀) × (1...𝑁)))))
30 opex 4853 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpt2i 7106 . . . . . . . . . . 11 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32 elpreima 6230 . . . . . . . . . . 11 ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ) → (𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
3331, 32ax-mp 5 . . . . . . . . . 10 (𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))))
3433a1i 11 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
35 simplr 787 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3635fveq2d 6092 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨(1st𝑥), (2nd𝑥)⟩))
37 df-ov 6530 . . . . . . . . . . . . . . . . . 18 ((1st𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd𝑥)) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨(1st𝑥), (2nd𝑥)⟩)
3836, 37syl6eqr 2661 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ((1st𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd𝑥)))
39 breq1 4580 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 < 𝐾 ↔ (1st𝑥) < 𝐾))
40 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → 𝑖 = (1st𝑥))
41 oveq1 6534 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 + 1) = ((1st𝑥) + 1))
4239, 40, 41ifbieq12d 4062 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)))
4342opeq1d 4340 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (1st𝑥) → ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
44 breq1 4580 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 < 𝐿 ↔ (2nd𝑥) < 𝐿))
45 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → 𝑗 = (2nd𝑥))
46 oveq1 6534 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 + 1) = ((2nd𝑥) + 1))
4744, 45, 46ifbieq12d 4062 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)))
4847opeq2d 4341 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (2nd𝑥) → ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩)
49 opex 4853 . . . . . . . . . . . . . . . . . . 19 ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ V
5043, 48, 5, 49ovmpt2 6672 . . . . . . . . . . . . . . . . . 18 (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) → ((1st𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd𝑥)) = ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩)
5150adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((1st𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd𝑥)) = ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩)
5238, 51eqtrd 2643 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩)
5352eleq1d 2671 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁))))
54 opelxp 5060 . . . . . . . . . . . . . . 15 (⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁)))
5553, 54syl6bb 274 . . . . . . . . . . . . . 14 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁))))
56 ifel 4078 . . . . . . . . . . . . . . . 16 (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))))
57 simplrl 795 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℕ)
5857nnred 10885 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℝ)
5913nnred 10885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 ∈ ℝ)
6059ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ ℕ)
6261nnred 10885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
6362ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝐾)
6558, 60, 64ltled 10037 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝐾)
66 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (1...𝑀) → 𝐾𝑀)
6712, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾𝑀)
6867ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾𝑀)
6958, 60, 63, 65, 68letrd 10046 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝑀)
7057, 69jca 552 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀))
7161nnzd 11316 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℤ)
72 fznn 12236 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7473ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7570, 74mpbird 245 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ (1...𝑀))
7658, 60, 63, 64, 68ltletrd 10049 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝑀)
7761ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78 nnltlem1 11279 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
7957, 77, 78syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
8076, 79mpbid 220 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ (𝑀 − 1))
8175, 802thd 253 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
8281pm5.32da 670 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
83 fznn 12236 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℤ → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8471, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8584ad2antrr 757 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
86 simprl 789 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℕ)
8786peano2nnd 10887 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((1st𝑥) + 1) ∈ ℕ)
8887biantrurd 527 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8986nnzd 11316 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
9071ad2antrr 757 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91 zltp1le 11263 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ ((1st𝑥) + 1) ≤ 𝑀))
92 zltlem1 11266 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9391, 92bitr3d 268 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9489, 90, 93syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9585, 88, 943bitr2d 294 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
9695anbi2d 735 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
9782, 96orbi12d 741 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))))
98 pm4.42 994 . . . . . . . . . . . . . . . . . 18 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)))
99 ancom 464 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
100 ancom 464 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
10199, 100orbi12i 541 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10298, 101bitri 262 . . . . . . . . . . . . . . . . 17 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10397, 102syl6bbr 276 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))) ↔ (1st𝑥) ≤ (𝑀 − 1)))
10456, 103syl5bb 270 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
105 ifel 4078 . . . . . . . . . . . . . . . 16 (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))))
106 simplrr 796 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℕ)
107106nnred 10885 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℝ)
10816nnred 10885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ ℝ)
109108ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℕ)
111110nnred 10885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℝ)
112111ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝐿)
114107, 109, 113ltled 10037 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝐿)
115 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ (1...𝑁) → 𝐿𝑁)
11615, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿𝑁)
117116ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿𝑁)
118107, 109, 112, 114, 117letrd 10046 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝑁)
119106, 118jca 552 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁))
120110nnzd 11316 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
121 fznn 12236 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℤ → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
123122ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
124119, 123mpbird 245 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ (1...𝑁))
125107, 109, 112, 113, 117ltletrd 10049 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝑁)
126110ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127 nnltlem1 11279 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
128106, 126, 127syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
129125, 128mpbid 220 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ (𝑁 − 1))
130124, 1292thd 253 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
131130pm5.32da 670 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
132 fznn 12236 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℤ → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
133120, 132syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
134133ad2antrr 757 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
135 simprr 791 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
136135peano2nnd 10887 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((2nd𝑥) + 1) ∈ ℕ)
137136biantrurd 527 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
138135nnzd 11316 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℤ)
139120ad2antrr 757 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140 zltp1le 11263 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ ((2nd𝑥) + 1) ≤ 𝑁))
141 zltlem1 11266 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
142140, 141bitr3d 268 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
143138, 139, 142syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
144134, 137, 1433bitr2d 294 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
145144anbi2d 735 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
146131, 145orbi12d 741 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
147 pm4.42 994 . . . . . . . . . . . . . . . . . 18 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)))
148 ancom 464 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
149 ancom 464 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
150148, 149orbi12i 541 . . . . . . . . . . . . . . . . . 18 ((((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
151147, 150bitri 262 . . . . . . . . . . . . . . . . 17 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
152146, 151syl6bbr 276 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
153105, 152syl5bb 270 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
154104, 153anbi12d 742 . . . . . . . . . . . . . 14 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁)) ↔ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
15555, 154bitrd 266 . . . . . . . . . . . . 13 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
156155pm5.32da 670 . . . . . . . . . . . 12 ((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) → ((((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
157 1zzd 11244 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ ℤ)
15871, 157zsubcld 11322 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℤ)
159 fznn 12236 . . . . . . . . . . . . . . . 16 ((𝑀 − 1) ∈ ℤ → ((1st𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1))))
160158, 159syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((1st𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1))))
161120, 157zsubcld 11322 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
162 fznn 12236 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ ℤ → ((2nd𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))))
163161, 162syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((2nd𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))))
164160, 163anbi12d 742 . . . . . . . . . . . . . 14 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
165 an4 860 . . . . . . . . . . . . . 14 ((((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
166164, 165syl6bb 274 . . . . . . . . . . . . 13 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
167166adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
168156, 167bitr4d 269 . . . . . . . . . . 11 ((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) → ((((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1)))))
169168pm5.32da 670 . . . . . . . . . 10 (𝜑 → ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))))))
170 elxp6 7069 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)))
171170anbi1i 726 . . . . . . . . . . 11 ((𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ ((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))))
172 anass 678 . . . . . . . . . . 11 (((𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
173171, 172bitri 262 . . . . . . . . . 10 ((𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
174 elxp6 7069 . . . . . . . . . 10 (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1)))))
175169, 173, 1743bitr4g 301 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ (ℕ × ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ 𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
17629, 34, 1753bitrd 292 . . . . . . . 8 (𝜑 → (𝑥 ∈ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ dom 𝐴) ↔ 𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
177176eqrdv 2607 . . . . . . 7 (𝜑 → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) “ dom 𝐴) = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
17825, 177syl5eq 2655 . . . . . 6 (𝜑 → dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
17924, 178eqtrd 2643 . . . . 5 (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180179feq2d 5930 . . . 4 (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
18123, 180mpbid 220 . . 3 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
18219rneqd 5261 . . . . 5 (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5294 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184182, 183syl6eqss 3617 . . . 4 (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185 frn 5952 . . . . 5 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴𝐵)
1861, 2, 1853syl 18 . . . 4 (𝜑 → ran 𝐴𝐵)
187184, 186sstrd 3577 . . 3 (𝜑 → ran 𝑆𝐵)
188 fss 5955 . . 3 ((𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆 ∧ ran 𝑆𝐵) → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
189181, 187, 188syl2anc 690 . 2 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190 reldmmap 7731 . . . . . 6 Rel dom ↑𝑚
191190ovrcl 6562 . . . . 5 (𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))) → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
1921, 191syl 17 . . . 4 (𝜑 → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
193192simpld 473 . . 3 (𝜑𝐵 ∈ V)
194 ovex 6555 . . . 4 (1...(𝑀 − 1)) ∈ V
195 ovex 6555 . . . 4 (1...(𝑁 − 1)) ∈ V
196194, 195xpex 6838 . . 3 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197 elmapg 7735 . . 3 ((𝐵 ∈ V ∧ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V) → (𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
198193, 196, 197sylancl 692 . 2 (𝜑 → (𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199189, 198mpbird 245 1 (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  ifcif 4035  cop 4130   class class class wbr 4577   × cxp 5026  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  ccom 5032  Fun wfun 5784   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7035  2nd c2nd 7036  𝑚 cmap 7722  cr 9792  1c1 9794   + caddc 9796   < clt 9931  cle 9932  cmin 10118  cn 10870  cz 11213  ...cfz 12155  subMat1csmat 29021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-smat 29022
This theorem is referenced by:  smatcl  29030  1smat1  29032
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