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Theorem smatrcl 33757
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 8888 . . . . . . . 8 (𝐴 ∈ (𝐵m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3 ffun 6740 . . . . . . . 8 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (𝜑 → Fun 𝐴)
5 eqid 2735 . . . . . . . . 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 6608 . . . . . . 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 13592 . . . . . . . . . 10 (1...𝑀) ⊆ ℕ
12 smat.k . . . . . . . . . 10 (𝜑𝐾 ∈ (1...𝑀))
1311, 12sselid 3993 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ)
14 fz1ssnn 13592 . . . . . . . . . 10 (1...𝑁) ⊆ ℕ
15 smat.l . . . . . . . . . 10 (𝜑𝐿 ∈ (1...𝑁))
1614, 15sselid 3993 . . . . . . . . 9 (𝜑𝐿 ∈ ℕ)
17 smatfval 33756 . . . . . . . . 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 1370 . . . . . . . 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 2787 . . . . . . 7 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 6590 . . . . . 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 6768 . . . . 5 (Fun 𝑆𝑆:dom 𝑆⟶ran 𝑆)
2321, 22sylib 218 . . . 4 (𝜑𝑆:dom 𝑆⟶ran 𝑆)
2419dmeqd 5919 . . . . . 6 (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 6276 . . . . . . 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 6746 . . . . . . . . . . . 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 6080 . . . . . . . . . 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 5475 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpoi 8094 . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . 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 4559 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st𝑥) < 𝐾, (1st𝑥), ((1st𝑥) + 1)))
4342opeq1d 4884 . . . . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . 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 4559 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd𝑥) < 𝐿, (2nd𝑥), ((2nd𝑥) + 1)))
4847opeq2d 4885 . . . . . . . . . . . . . . . . . . 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 5475 . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . 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 5725 . . . . . . . . . . . . . . 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 4575 . . . . . . . . . . . . . . . 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 12279 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ∈ ℝ)
5913nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐾 ∈ ℝ)
6059ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ ℕ)
6261nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . 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 11407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (1st𝑥) < 𝐾) → (1st𝑥) ≤ 𝐾)
66 elfzle2 13565 . . . . . . . . . . . . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . . . . . . . . . . . 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 12638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℤ)
72 fznn 13629 . . . . . . . . . . . . . . . . . . . . . . 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 11419 . . . . . . . . . . . . . . . . . . . . 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 12683 . . . . . . . . . . . . . . . . . . . . . 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 13629 . . . . . . . . . . . . . . . . . . . . . 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 12281 . . . . . . . . . . . . . . . . . . . . 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 12638 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (1st𝑥) ∈ ℤ)
9071ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91 zltp1le 12665 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st𝑥) < 𝑀 ↔ ((1st𝑥) + 1) ≤ 𝑀))
92 zltlem1 12668 . . . . . . . . . . . . . . . . . . . . . 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 918 . . . . . . . . . . . . . . . . 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 1053 . . . . . . . . . . . . . . . . . 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 914 . . . . . . . . . . . . . . . . . 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 4575 . . . . . . . . . . . . . . . 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 12279 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ∈ ℝ)
10816nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐿 ∈ ℝ)
109108ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℕ)
111110nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . 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 11407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) ∧ (2nd𝑥) < 𝐿) → (2nd𝑥) ≤ 𝐿)
115 elfzle2 13565 . . . . . . . . . . . . . . . . . . . . . . . . 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 11416 . . . . . . . . . . . . . . . . . . . . . 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 12638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
121 fznn 13629 . . . . . . . . . . . . . . . . . . . . . . 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 11419 . . . . . . . . . . . . . . . . . . . . 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 12683 . . . . . . . . . . . . . . . . . . . . . 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 13629 . . . . . . . . . . . . . . . . . . . . . 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 12281 . . . . . . . . . . . . . . . . . . . . 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 12638 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → (2nd𝑥) ∈ ℤ)
139120ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩) ∧ ((1st𝑥) ∈ ℕ ∧ (2nd𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140 zltp1le 12665 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑥) < 𝑁 ↔ ((2nd𝑥) + 1) ≤ 𝑁))
141 zltlem1 12668 . . . . . . . . . . . . . . . . . . . . . 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 918 . . . . . . . . . . . . . . . . 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 1053 . . . . . . . . . . . . . . . . . 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 914 . . . . . . . . . . . . . . . . . 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 12646 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ ℤ)
15871, 157zsubcld 12725 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 − 1) ∈ ℤ)
159 fznn 13629 . . . . . . . . . . . . . . . 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 12725 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
162 fznn 13629 . . . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . 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 8047 . . . . . . . . . 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 2733 . . . . . . 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 2787 . . . . . 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 2775 . . . . 5 (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180179feq2d 6723 . . . 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 5952 . . . . 5 (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5989 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184182, 183eqsstrdi 4050 . . . 4 (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185 frn 6744 . . . . 5 (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴𝐵)
1861, 2, 1853syl 18 . . . 4 (𝜑 → ran 𝐴𝐵)
187184, 186sstrd 4006 . . 3 (𝜑 → ran 𝑆𝐵)
188 fss 6753 . . 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 8874 . . . . . 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 7772 . . 3 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197 elmapg 8878 . . 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 847   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  ifcif 4531  cop 4637   class class class wbr 5148   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cr 11152  1c1 11154   + caddc 11156   < clt 11293  cle 11294  cmin 11490  cn 12264  cz 12611  ...cfz 13544  subMat1csmat 33754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-smat 33755
This theorem is referenced by:  smatcl  33763  1smat1  33765
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