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Theorem smatrcl 29190
 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 7765 . . . . . . . 8 (𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3 ffun 5961 . . . . . . . 8 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (𝜑 → Fun 𝐴)
5 eqid 2610 . . . . . . . . 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 6660 . . . . . . . 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 5842 . . . . . . 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 691 . . . . . 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 12243 . . . . . . . . . 10 (1...𝑀) ⊆ ℕ
12 smat.k . . . . . . . . . 10 (𝜑𝐾 ∈ (1...𝑀))
1311, 12sseldi 3566 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ)
14 fz1ssnn 12243 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
15 smat.l . . . . . . . . . 10 (𝜑𝐿 ∈ (1...𝑁))
1614, 15sseldi 3566 . . . . . . . . 9 (𝜑𝐿 ∈ ℕ)
17 smatfval 29189 . . . . . . . . 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 1318 . . . . . . . 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 2656 . . . . . . 7 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 5825 . . . . . 6 (𝜑 → (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
219, 20mpbird 246 . . . . 5 (𝜑 → Fun 𝑆)
22 fdmrn 5977 . . . . 5 (Fun 𝑆𝑆:dom 𝑆⟶ran 𝑆)
2321, 22sylib 207 . . . 4 (𝜑𝑆:dom 𝑆⟶ran 𝑆)
2419dmeqd 5248 . . . . . 6 (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 5560 . . . . . . 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 5964 . . . . . . . . . . . 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 5385 . . . . . . . . . 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 2673 . . . . . . . . 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 4859 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpt2i 7128 . . . . . . . . . . 11 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32 elpreima 6245 . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3635fveq2d 6107 . . . . . . . . . . . . . . . . . 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 6552 . . . . . . . . . . . . . . . . . 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 2662 . . . . . . . . . . . . . . . . 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 4586 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 < 𝐾 ↔ (1st𝑥) < 𝐾))
40 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → 𝑖 = (1st𝑥))
41 oveq1 6556 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st𝑥) → (𝑖 + 1) = ((1st𝑥) + 1))
4239, 40, 41ifbieq12d 4063 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)))
4342opeq1d 4346 . . . . . . . . . . . . . . . . . . 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 4586 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 < 𝐿 ↔ (2nd𝑥) < 𝐿))
45 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → 𝑗 = (2nd𝑥))
46 oveq1 6556 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑥) → (𝑗 + 1) = ((2nd𝑥) + 1))
4744, 45, 46ifbieq12d 4063 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)))
4847opeq2d 4347 . . . . . . . . . . . . . . . . . . 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 4859 . . . . . . . . . . . . . . . . . . 19 ⟨if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)), if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1))⟩ ∈ V
5043, 48, 5, 49ovmpt2 6694 . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . . . 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 5070 . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . 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 4079 . . . . . . . . . . . . . . . 16 (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))))
57 simplrl 796 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℕ)
5857nnred 10912 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℝ)
5913nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 ∈ ℝ)
6059ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ ℕ)
6261nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
6362ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝐾)
6558, 60, 64ltled 10064 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝐾)
66 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (1...𝑀) → 𝐾𝑀)
6712, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾𝑀)
6867ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾𝑀)
6958, 60, 63, 65, 68letrd 10073 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝑀)
7057, 69jca 553 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀))
7161nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℤ)
72 fznn 12278 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℤ → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7473ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ ((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ 𝑀)))
7570, 74mpbird 246 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ (1...𝑀))
7658, 60, 63, 64, 68ltletrd 10076 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) < 𝑀)
7761ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78 nnltlem1 11320 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
7957, 77, 78syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
8076, 79mpbid 221 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ (𝑀 − 1))
8175, 802thd 254 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → ((1st𝑥) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
8281pm5.32da 671 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ↔ ((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
83 fznn 12278 . . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
86 simprl 790 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℕ)
8786peano2nnd 10914 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((1st𝑥) + 1) ∈ ℕ)
8887biantrurd 528 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (((1st𝑥) + 1) ∈ ℕ ∧ ((1st𝑥) + 1) ≤ 𝑀)))
8986nnzd 11357 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
9071ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91 zltp1le 11304 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ ((1st𝑥) + 1) ≤ 𝑀))
92 zltlem1 11307 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9391, 92bitr3d 269 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9489, 90, 93syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ≤ 𝑀 ↔ (1st𝑥) ≤ (𝑀 − 1)))
9585, 88, 943bitr2d 295 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((1st𝑥) + 1) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
9695anbi2d 736 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
9782, 96orbi12d 742 . . . . . . . . . . . . . . . . 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 pm4.42 995 . . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) ≤ (𝑀 − 1) ∧ (1st𝑥) < 𝐾) ∨ ((1st𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st𝑥) < 𝐾)) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10298, 101bitri 263 . . . . . . . . . . . . . . . . 17 ((1st𝑥) ≤ (𝑀 − 1) ↔ (((1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st𝑥) < 𝐾 ∧ (1st𝑥) ≤ (𝑀 − 1))))
10397, 102syl6bbr 277 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((1st𝑥) < 𝐾 ∧ (1st𝑥) ∈ (1...𝑀)) ∨ (¬ (1st𝑥) < 𝐾 ∧ ((1st𝑥) + 1) ∈ (1...𝑀))) ↔ (1st𝑥) ≤ (𝑀 − 1)))
10456, 103syl5bb 271 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)) ∈ (1...𝑀) ↔ (1st𝑥) ≤ (𝑀 − 1)))
105 ifel 4079 . . . . . . . . . . . . . . . 16 (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))))
106 simplrr 797 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℕ)
107106nnred 10912 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℝ)
10816nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ ℝ)
109108ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℕ)
111110nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℝ)
112111ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝐿)
114107, 109, 113ltled 10064 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝐿)
115 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ (1...𝑁) → 𝐿𝑁)
11615, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿𝑁)
117116ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿𝑁)
118107, 109, 112, 114, 117letrd 10073 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝑁)
119106, 118jca 553 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁))
120110nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
121 fznn 12278 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℤ → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
123122ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ 𝑁)))
124119, 123mpbird 246 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ (1...𝑁))
125107, 109, 112, 113, 117ltletrd 10076 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) < 𝑁)
126110ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127 nnltlem1 11320 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
128106, 126, 127syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
129125, 128mpbid 221 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ (𝑁 − 1))
130124, 1292thd 254 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → ((2nd𝑥) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
131130pm5.32da 671 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ↔ ((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
132 fznn 12278 . . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
135 simprr 792 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℕ)
136135peano2nnd 10914 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((2nd𝑥) + 1) ∈ ℕ)
137136biantrurd 528 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (((2nd𝑥) + 1) ∈ ℕ ∧ ((2nd𝑥) + 1) ≤ 𝑁)))
138135nnzd 11357 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℤ)
139120ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140 zltp1le 11304 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ ((2nd𝑥) + 1) ≤ 𝑁))
141 zltlem1 11307 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
142140, 141bitr3d 269 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
143138, 139, 142syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ≤ 𝑁 ↔ (2nd𝑥) ≤ (𝑁 − 1)))
144134, 137, 1433bitr2d 295 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (((2nd𝑥) + 1) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
145144anbi2d 736 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
146131, 145orbi12d 742 . . . . . . . . . . . . . . . . 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 pm4.42 995 . . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . . . 18 ((((2nd𝑥) ≤ (𝑁 − 1) ∧ (2nd𝑥) < 𝐿) ∨ ((2nd𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd𝑥) < 𝐿)) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
151147, 150bitri 263 . . . . . . . . . . . . . . . . 17 ((2nd𝑥) ≤ (𝑁 − 1) ↔ (((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ (2nd𝑥) ≤ (𝑁 − 1))))
152146, 151syl6bbr 277 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → ((((2nd𝑥) < 𝐿 ∧ (2nd𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd𝑥) < 𝐿 ∧ ((2nd𝑥) + 1) ∈ (1...𝑁))) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
153105, 152syl5bb 271 . . . . . . . . . . . . . . 15 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)) ∈ (1...𝑁) ↔ (2nd𝑥) ≤ (𝑁 − 1)))
154104, 153anbi12d 743 . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . 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 671 . . . . . . . . . . . 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 11285 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ ℤ)
15871, 157zsubcld 11363 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℤ)
159 fznn 12278 . . . . . . . . . . . . . . . 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 11363 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
162 fznn 12278 . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . 14 (𝜑 → (((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1)))))
165 an4 861 . . . . . . . . . . . . . 14 ((((1st𝑥) ∈ ℕ ∧ (1st𝑥) ≤ (𝑀 − 1)) ∧ ((2nd𝑥) ∈ ℕ ∧ (2nd𝑥) ≤ (𝑁 − 1))) ↔ (((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ) ∧ ((1st𝑥) ≤ (𝑀 − 1) ∧ (2nd𝑥) ≤ (𝑁 − 1))))
166164, 165syl6bb 275 . . . . . . . . . . . . 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 270 . . . . . . . . . . 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 671 . . . . . . . . . 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 7091 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)))
171170anbi1i 727 . . . . . . . . . . 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 679 . . . . . . . . . . 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 263 . . . . . . . . . 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 7091 . . . . . . . . . 10 (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd𝑥) ∈ (1...(𝑁 − 1)))))
175169, 173, 1743bitr4g 302 . . . . . . . . 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 293 . . . . . . . 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 2608 . . . . . . 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 2656 . . . . . 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 2644 . . . . 5 (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180179feq2d 5944 . . . 4 (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
18123, 180mpbid 221 . . 3 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
18219rneqd 5274 . . . . 5 (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5307 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184182, 183syl6eqss 3618 . . . 4 (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185 frn 5966 . . . . 5 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴𝐵)
1861, 2, 1853syl 18 . . . 4 (𝜑 → ran 𝐴𝐵)
187184, 186sstrd 3578 . . 3 (𝜑 → ran 𝑆𝐵)
188 fss 5969 . . 3 ((𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆 ∧ ran 𝑆𝐵) → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
189181, 187, 188syl2anc 691 . 2 (𝜑𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190 reldmmap 7753 . . . . . 6 Rel dom ↑𝑚
191190ovrcl 6584 . . . . 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 474 . . 3 (𝜑𝐵 ∈ V)
194 ovex 6577 . . . 4 (1...(𝑀 − 1)) ∈ V
195 ovex 6577 . . . 4 (1...(𝑁 − 1)) ∈ V
196194, 195xpex 6860 . . 3 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197 elmapg 7757 . . 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 693 . 2 (𝜑 → (𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199189, 198mpbird 246 1 (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ifcif 4036  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℤcz 11254  ...cfz 12197  subMat1csmat 29187 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-smat 29188 This theorem is referenced by:  smatcl  29196  1smat1  29198
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