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Theorem smatrcl 30192
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 8047 . . . . . . . 8 (𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3 ffun 6209 . . . . . . . 8 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (𝜑 → Fun 𝐴)
5 eqid 2760 . . . . . . . . 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 6928 . . . . . . . 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 6089 . . . . . . 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 696 . . . . . 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 12585 . . . . . . . . . 10 (1...𝑀) ⊆ ℕ
12 smat.k . . . . . . . . . 10 (𝜑𝐾 ∈ (1...𝑀))
1311, 12sseldi 3742 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ)
14 fz1ssnn 12585 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
15 smat.l . . . . . . . . . 10 (𝜑𝐿 ∈ (1...𝑁))
1614, 15sseldi 3742 . . . . . . . . 9 (𝜑𝐿 ∈ ℕ)
17 smatfval 30191 . . . . . . . . 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 1477 . . . . . . . 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 2806 . . . . . . 7 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 6071 . . . . . 6 (𝜑 → (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
219, 20mpbird 247 . . . . 5 (𝜑 → Fun 𝑆)
22 fdmrn 6225 . . . . 5 (Fun 𝑆𝑆:dom 𝑆⟶ran 𝑆)
2321, 22sylib 208 . . . 4 (𝜑𝑆:dom 𝑆⟶ran 𝑆)
2419dmeqd 5481 . . . . . 6 (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 5804 . . . . . . 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 6212 . . . . . . . . . . . 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 5624 . . . . . . . . . 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 2825 . . . . . . . . 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 5081 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpt2i 7408 . . . . . . . . . . 11 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32 elpreima 6501 . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3635fveq2d 6357 . . . . . . . . . . . . . . . . . 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 6817 . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 < 𝐾 ↔ (1st𝑥) < 𝐾))
40 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → 𝑖 = (1st𝑥))
41 oveq1 6821 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 + 1) = ((1st𝑥) + 1))
4239, 40, 41ifbieq12d 4257 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)))
4342opeq1d 4559 . . . . . . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 < 𝐿 ↔ (2nd𝑥) < 𝐿))
45 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → 𝑗 = (2nd𝑥))
46 oveq1 6821 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 + 1) = ((2nd𝑥) + 1))
4744, 45, 46ifbieq12d 4257 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)))
4847opeq2d 4560 . . . . . . . . . . . . . . . . . . 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 5081 . . . . . . . . . . . . . . . . . . 19 ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ V
5043, 48, 5, 49ovmpt2 6962 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 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 5303 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . . . . 16 (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))))
57 simplrl 819 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℕ)
5857nnred 11247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℝ)
5913nnred 11247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 ∈ ℝ)
6059ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ ℕ)
6261nnred 11247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
6362ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝐾)
6558, 60, 64ltled 10397 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝐾)
66 elfzle2 12558 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (1...𝑀) → 𝐾𝑀)
6712, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾𝑀)
6867ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾𝑀)
6958, 60, 63, 65, 68letrd 10406 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝑀)
7057, 69jca 555 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀))
7161nnzd 11693 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℤ)
72 fznn 12621 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7473ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7570, 74mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ (1...𝑀))
7658, 60, 63, 64, 68ltletrd 10409 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝑀)
7761ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78 nnltlem1 11656 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
7957, 77, 78syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
8076, 79mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ (𝑀 − 1))
8175, 802thd 255 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
8281pm5.32da 676 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
83 fznn 12621 . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
86 simprl 811 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℕ)
8786peano2nnd 11249 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((1st𝑥) + 1) ∈ ℕ)
8887biantrurd 530 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8986nnzd 11693 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
9071ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91 zltp1le 11639 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ ((1st𝑥) + 1) ≤ 𝑀))
92 zltlem1 11642 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9391, 92bitr3d 270 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9489, 90, 93syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9585, 88, 943bitr2d 296 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
9695anbi2d 742 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
9782, 96orbi12d 748 . . . . . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . . . . . . 18 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)))
99 ancom 465 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
100 ancom 465 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)))
10199, 100orbi12i 544 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10298, 101bitri 264 . . . . . . . . . . . . . . . . 17 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10397, 102syl6bbr 278 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))) ↔ (1st𝑥) ≤ (𝑀 − 1)))
10456, 103syl5bb 272 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
105 ifel 4273 . . . . . . . . . . . . . . . 16 (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))))
106 simplrr 820 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℕ)
107106nnred 11247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℝ)
10816nnred 11247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ ℝ)
109108ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℕ)
111110nnred 11247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℝ)
112111ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝐿)
114107, 109, 113ltled 10397 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝐿)
115 elfzle2 12558 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ (1...𝑁) → 𝐿𝑁)
11615, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿𝑁)
117116ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿𝑁)
118107, 109, 112, 114, 117letrd 10406 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝑁)
119106, 118jca 555 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁))
120110nnzd 11693 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
121 fznn 12621 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℤ → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
123122ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
124119, 123mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ (1...𝑁))
125107, 109, 112, 113, 117ltletrd 10409 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝑁)
126110ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127 nnltlem1 11656 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
128106, 126, 127syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
129125, 128mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ (𝑁 − 1))
130124, 1292thd 255 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
131130pm5.32da 676 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
132 fznn 12621 . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
135 simprr 813 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
136135peano2nnd 11249 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((2nd𝑥) + 1) ∈ ℕ)
137136biantrurd 530 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
138135nnzd 11693 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℤ)
139120ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140 zltp1le 11639 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ ((2nd𝑥) + 1) ≤ 𝑁))
141 zltlem1 11642 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
142140, 141bitr3d 270 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
143138, 139, 142syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
144134, 137, 1433bitr2d 296 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
145144anbi2d 742 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
146131, 145orbi12d 748 . . . . . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . . . . . . 18 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)))
148 ancom 465 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
149 ancom 465 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)))
150148, 149orbi12i 544 . . . . . . . . . . . . . . . . . 18 ((((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
151147, 150bitri 264 . . . . . . . . . . . . . . . . 17 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
152146, 151syl6bbr 278 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
153105, 152syl5bb 272 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
154104, 153anbi12d 749 . . . . . . . . . . . . . 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 268 . . . . . . . . . . . . 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 676 . . . . . . . . . . . 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 11620 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ ℤ)
15871, 157zsubcld 11699 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℤ)
159 fznn 12621 . . . . . . . . . . . . . . . 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 11699 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
162 fznn 12621 . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . 14 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
165 an4 900 . . . . . . . . . . . . . 14 ((((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
166164, 165syl6bb 276 . . . . . . . . . . . . 13 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
167166adantr 472 . . . . . . . . . . . 12 ((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
168156, 167bitr4d 271 . . . . . . . . . . 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 676 . . . . . . . . . 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 7368 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)))
171170anbi1i 733 . . . . . . . . . . 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 684 . . . . . . . . . . 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 264 . . . . . . . . . 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 7368 . . . . . . . . . 10 (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1)))))
175169, 173, 1743bitr4g 303 . . . . . . . . 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 294 . . . . . . . 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 2758 . . . . . . 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 2806 . . . . . 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 2794 . . . . 5 (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180179feq2d 6192 . . . 4 (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
18123, 180mpbid 222 . . 3 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
18219rneqd 5508 . . . . 5 (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5541 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184182, 183syl6eqss 3796 . . . 4 (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185 frn 6214 . . . . 5 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴𝐵)
1861, 2, 1853syl 18 . . . 4 (𝜑 → ran 𝐴𝐵)
187184, 186sstrd 3754 . . 3 (𝜑 → ran 𝑆𝐵)
188 fss 6217 . . 3 ((𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆 ∧ ran 𝑆𝐵) → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
189181, 187, 188syl2anc 696 . 2 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190 reldmmap 8034 . . . . . 6 Rel dom ↑𝑚
191190ovrcl 6850 . . . . 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 477 . . 3 (𝜑𝐵 ∈ V)
194 ovex 6842 . . . 4 (1...(𝑀 − 1)) ∈ V
195 ovex 6842 . . . 4 (1...(𝑁 − 1)) ∈ V
196194, 195xpex 7128 . . 3 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197 elmapg 8038 . . 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 697 . 2 (𝜑 → (𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199189, 198mpbird 247 1 (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  ifcif 4230  cop 4327   class class class wbr 4804   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332  2nd c2nd 7333  𝑚 cmap 8025  cr 10147  1c1 10149   + caddc 10151   < clt 10286  cle 10287  cmin 10478  cn 11232  cz 11589  ...cfz 12539  subMat1csmat 30189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-smat 30190
This theorem is referenced by:  smatcl  30198  1smat1  30200
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