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Theorem smatrcl 33795
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 8889 . . . . . . . 8 (𝐴 ∈ (𝐵m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3 ffun 6739 . . . . . . . 8 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (𝜑 → Fun 𝐴)
5 eqid 2737 . . . . . . . . 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))⟩)
65mpofun 7557 . . . . . . . 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 6606 . . . . . . 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 584 . . . . . 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 13595 . . . . . . . . . 10 (1...𝑀) ⊆ ℕ
12 smat.k . . . . . . . . . 10 (𝜑𝐾 ∈ (1...𝑀))
1311, 12sselid 3981 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ)
14 fz1ssnn 13595 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
15 smat.l . . . . . . . . . 10 (𝜑𝐿 ∈ (1...𝑁))
1614, 15sselid 3981 . . . . . . . . 9 (𝜑𝐿 ∈ ℕ)
17 smatfval 33794 . . . . . . . . 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 1373 . . . . . . . 8 (𝜑 → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1910, 18eqtrid 2789 . . . . . . 7 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 6588 . . . . . 6 (𝜑 → (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
219, 20mpbird 257 . . . . 5 (𝜑 → Fun 𝑆)
22 fdmrn 6767 . . . . 5 (Fun 𝑆𝑆:dom 𝑆⟶ran 𝑆)
2321, 22sylib 218 . . . 4 (𝜑𝑆:dom 𝑆⟶ran 𝑆)
2419dmeqd 5916 . . . . . 6 (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 6274 . . . . . . 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 6745 . . . . . . . . . . . 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 6078 . . . . . . . . . 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 2827 . . . . . . . . 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 5469 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpoi 8095 . . . . . . . . . . 11 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32 elpreima 7078 . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3635fveq2d 6910 . . . . . . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . . 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, 37eqtr4di 2795 . . . . . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 < 𝐾 ↔ (1st𝑥) < 𝐾))
40 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → 𝑖 = (1st𝑥))
41 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 + 1) = ((1st𝑥) + 1))
4239, 40, 41ifbieq12d 4554 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)))
4342opeq1d 4879 . . . . . . . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 < 𝐿 ↔ (2nd𝑥) < 𝐿))
45 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → 𝑗 = (2nd𝑥))
46 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 + 1) = ((2nd𝑥) + 1))
4744, 45, 46ifbieq12d 4554 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)))
4847opeq2d 4880 . . . . . . . . . . . . . . . . . . 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 5469 . . . . . . . . . . . . . . . . . . 19 ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ V
5043, 48, 5, 49ovmpo 7593 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 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 5721 . . . . . . . . . . . . . . 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, 54bitrdi 287 . . . . . . . . . . . . . 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 4570 . . . . . . . . . . . . . . . 16 (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))))
57 simplrl 777 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℕ)
5857nnred 12281 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℝ)
5913nnred 12281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 ∈ ℝ)
6059ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ ℕ)
6261nnred 12281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
6362ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝐾)
6558, 60, 64ltled 11409 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝐾)
66 elfzle2 13568 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (1...𝑀) → 𝐾𝑀)
6712, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾𝑀)
6867ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾𝑀)
6958, 60, 63, 65, 68letrd 11418 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝑀)
7057, 69jca 511 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀))
7161nnzd 12640 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℤ)
72 fznn 13632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7473ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7570, 74mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ (1...𝑀))
7658, 60, 63, 64, 68ltletrd 11421 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝑀)
7761ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78 nnltlem1 12685 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
7957, 77, 78syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
8076, 79mpbid 232 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ (𝑀 − 1))
8175, 802thd 265 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
8281pm5.32da 579 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
83 fznn 13632 . . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
86 simprl 771 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℕ)
8786peano2nnd 12283 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((1st𝑥) + 1) ∈ ℕ)
8887biantrurd 532 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8986nnzd 12640 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
9071ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91 zltp1le 12667 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ ((1st𝑥) + 1) ≤ 𝑀))
92 zltlem1 12670 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9391, 92bitr3d 281 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9489, 90, 93syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9585, 88, 943bitr2d 307 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
9695anbi2d 630 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
9782, 96orbi12d 919 . . . . . . . . . . . . . . . . 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 1054 . . . . . . . . . . . . . . . . . 18 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)))
99 ancom 460 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
100 ancom 460 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
10199, 100orbi12i 915 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10298, 101bitri 275 . . . . . . . . . . . . . . . . 17 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10397, 102bitr4di 289 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))) ↔ (1st𝑥) ≤ (𝑀 − 1)))
10456, 103bitrid 283 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
105 ifel 4570 . . . . . . . . . . . . . . . 16 (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))))
106 simplrr 778 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℕ)
107106nnred 12281 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℝ)
10816nnred 12281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ ℝ)
109108ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℕ)
111110nnred 12281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℝ)
112111ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝐿)
114107, 109, 113ltled 11409 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝐿)
115 elfzle2 13568 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ (1...𝑁) → 𝐿𝑁)
11615, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿𝑁)
117116ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿𝑁)
118107, 109, 112, 114, 117letrd 11418 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝑁)
119106, 118jca 511 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁))
120110nnzd 12640 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
121 fznn 13632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℤ → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
123122ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
124119, 123mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ (1...𝑁))
125107, 109, 112, 113, 117ltletrd 11421 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝑁)
126110ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127 nnltlem1 12685 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
128106, 126, 127syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
129125, 128mpbid 232 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ (𝑁 − 1))
130124, 1292thd 265 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
131130pm5.32da 579 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
132 fznn 13632 . . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
135 simprr 773 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
136135peano2nnd 12283 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((2nd𝑥) + 1) ∈ ℕ)
137136biantrurd 532 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
138135nnzd 12640 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℤ)
139120ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140 zltp1le 12667 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ ((2nd𝑥) + 1) ≤ 𝑁))
141 zltlem1 12670 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
142140, 141bitr3d 281 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
143138, 139, 142syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
144134, 137, 1433bitr2d 307 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
145144anbi2d 630 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
146131, 145orbi12d 919 . . . . . . . . . . . . . . . . 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 1054 . . . . . . . . . . . . . . . . . 18 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)))
148 ancom 460 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
149 ancom 460 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
150148, 149orbi12i 915 . . . . . . . . . . . . . . . . . 18 ((((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
151147, 150bitri 275 . . . . . . . . . . . . . . . . 17 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
152146, 151bitr4di 289 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
153105, 152bitrid 283 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
154104, 153anbi12d 632 . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . 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 579 . . . . . . . . . . . 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 12648 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ ℤ)
15871, 157zsubcld 12727 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℤ)
159 fznn 13632 . . . . . . . . . . . . . . . 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 12727 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
162 fznn 13632 . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . 14 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
165 an4 656 . . . . . . . . . . . . . 14 ((((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
166164, 165bitrdi 287 . . . . . . . . . . . . 13 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
167166adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
168156, 167bitr4d 282 . . . . . . . . . . 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 579 . . . . . . . . . 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 8048 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)))
171170anbi1i 624 . . . . . . . . . . 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 468 . . . . . . . . . . 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 275 . . . . . . . . . 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 8048 . . . . . . . . . 10 (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1)))))
175169, 173, 1743bitr4g 314 . . . . . . . . 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 305 . . . . . . . 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 2735 . . . . . . 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, 177eqtrid 2789 . . . . . 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 2777 . . . . 5 (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180179feq2d 6722 . . . 4 (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
18123, 180mpbid 232 . . 3 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
18219rneqd 5949 . . . . 5 (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5986 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184182, 183eqsstrdi 4028 . . . 4 (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185 frn 6743 . . . . 5 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴𝐵)
1861, 2, 1853syl 18 . . . 4 (𝜑 → ran 𝐴𝐵)
187184, 186sstrd 3994 . . 3 (𝜑 → ran 𝑆𝐵)
188 fss 6752 . . 3 ((𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆 ∧ ran 𝑆𝐵) → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
189181, 187, 188syl2anc 584 . 2 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190 reldmmap 8875 . . . . . 6 Rel dom ↑m
191190ovrcl 7472 . . . . 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 494 . . 3 (𝜑𝐵 ∈ V)
194 ovex 7464 . . . 4 (1...(𝑀 − 1)) ∈ V
195 ovex 7464 . . . 4 (1...(𝑁 − 1)) ∈ V
196194, 195xpex 7773 . . 3 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197 elmapg 8879 . . 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 586 . 2 (𝜑 → (𝑆 ∈ (𝐵m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199189, 198mpbird 257 1 (𝜑𝑆 ∈ (𝐵m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  ifcif 4525  cop 4632   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  m cmap 8866  cr 11154  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cn 12266  cz 12613  ...cfz 13547  subMat1csmat 33792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-smat 33793
This theorem is referenced by:  smatcl  33801  1smat1  33803
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