Theorem smatrcl 34314

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	⊢ (𝜑 → 𝑀 ∈ ℕ)
smat.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
smat.k	⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
smat.l	⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
smat.a	⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))

Assertion

Ref	Expression
smatrcl	⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Proof of Theorem smatrcl

Dummy variables 𝑖 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smat.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
2		elmapi 8852	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3		ffun 6709	. . . . . . . 8 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
4	1, 2, 3	3syl 19	. . . . . . 7 ⊢ (𝜑 → Fun 𝐴)
5		eqid 2762	. . . . . . . . 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))⟩)
6	5	mpofun 7541	. . . . . . . 8 ⊢ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
8		funco 6577	. . . . . . 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))⟩)))
9	4, 7, 8	syl2anc 596	. . . . . 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 13614	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
12		smat.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
13	11, 12	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
14		fz1ssnn 13614	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
15		smat.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
16	14, 15	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℕ)
17		smatfval 34313	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝐴 ∈ (𝐵 ↑m ((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))⟩)))
18	13, 16, 1, 17	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
19	10, 18	eqtrid 2809	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
20	19	funeqd 6559	. . . . . 6 ⊢ (𝜑 → (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
21	9, 20	mpbird 260	. . . . 5 ⊢ (𝜑 → Fun 𝑆)
22		fdmrn 6738	. . . . 5 ⊢ (Fun 𝑆 ↔ 𝑆:dom 𝑆⟶ran 𝑆)
23	21, 22	sylib 221	. . . 4 ⊢ (𝜑 → 𝑆:dom 𝑆⟶ran 𝑆)
24	19	dmeqd 5893	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25		dmco 6255	. . . . . . 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 6716	. . . . . . . . . . . 12 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
27	1, 2, 26	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
28	27	imaeq2d 6060	. . . . . . . . . 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))))
29	28	eleq2d 2848	. . . . . . . . 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 5443	. . . . . . . . . . . 12 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
31	5, 30	fnmpoi 8071	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32		elpreima 7054	. . . . . . . . . . 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)))))
33	31, 32	ax-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))))
34	33	a1i 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 781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
36	35	fveq2d 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ 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))⟩)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
37		df-ov 7420	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd ‘𝑥)) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
38	36, 37	eqtr4di 2815	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ((1st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd ‘𝑥)))
39		breq1 5110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 < 𝐾 ↔ (1st ‘𝑥) < 𝐾))
40		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → 𝑖 = (1st ‘𝑥))
41		oveq1 7424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 + 1) = ((1st ‘𝑥) + 1))
42	39, 40, 41	ifbieq12d 4514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = (1st ‘𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)))
43	42	opeq1d 4842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (1st ‘𝑥) → ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
44		breq1 5110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 < 𝐿 ↔ (2nd ‘𝑥) < 𝐿))
45		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → 𝑗 = (2nd ‘𝑥))
46		oveq1 7424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 + 1) = ((2nd ‘𝑥) + 1))
47	44, 45, 46	ifbieq12d 4514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)))
48	47	opeq2d 4843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩)
49		opex 5443	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩ ∈ V
50	43, 48, 5, 49	ovmpo 7577	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) → ((1st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd ‘𝑥)) = ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩)
51	50	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((1st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd ‘𝑥)) = ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩)
52	38, 51	eqtrd 2797	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩)
53	52	eleq1d 2847	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁))))
54		opelxp 5695	. . . . . . . . . . . . . . 15 ⊢ (⟨if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)), if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)) ∈ (1...𝑁)))
55	53, 54	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)) ∈ (1...𝑁))))
56		ifel 4530	. . . . . . . . . . . . . . . 16 ⊢ (if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)) ∈ (1...𝑀) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ ((1st ‘𝑥) + 1) ∈ (1...𝑀))))
57		simplrl 789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℕ)
58	57	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℝ)
59	13	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ∈ ℝ)
60	59	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61		smat.m	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑀 ∈ ℕ)
62	61	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝐾)
65	58, 60, 64	ltled 11386	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝐾)
66		elfzle2 13586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1...𝑀) → 𝐾 ≤ 𝑀)
67	12, 66	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
68	67	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝐾 ≤ 𝑀)
69	58, 60, 63, 65, 68	letrd 11395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝑀)
70	57, 69	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀))
71	61	nnzd 12645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℤ)
72		fznn 13651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℤ → ((1st ‘𝑥) ∈ (1...𝑀) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀)))
73	71, 72	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1st ‘𝑥) ∈ (1...𝑀) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀)))
74	73	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → ((1st ‘𝑥) ∈ (1...𝑀) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀)))
75	70, 74	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ (1...𝑀))
76	58, 60, 63, 64, 68	ltletrd 11398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝑀)
77	61	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78		nnltlem1 12692	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
79	57, 77, 78	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
80	76, 79	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ (𝑀 − 1))
81	75, 80	2thd 268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → ((1st ‘𝑥) ∈ (1...𝑀) ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
82	81	pm5.32da 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ∈ (1...𝑀)) ↔ ((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
83		fznn 13651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℤ → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
84	71, 83	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
85	84	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
86		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℕ)
87	86	peano2nnd 12278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((1st ‘𝑥) + 1) ∈ ℕ)
88	87	biantrurd 542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((1st ‘𝑥) + 1) ≤ 𝑀 ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
89	86	nnzd 12645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℤ)
90	71	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91		zltp1le 12672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ ((1st ‘𝑥) + 1) ≤ 𝑀))
92		zltlem1 12675	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
93	91, 92	bitr3d 284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st ‘𝑥) + 1) ≤ 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
94	89, 90, 93	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((1st ‘𝑥) + 1) ≤ 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
95	85, 88, 94	3bitr2d 310	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
96	95	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((¬ (1st ‘𝑥) < 𝐾 ∧ ((1st ‘𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
97	82, 96	orbi12d 932	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ ((1st ‘𝑥) + 1) ∈ (1...𝑀))) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))))
98		pm4.42 1069	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ∨ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾)))
99		ancom 466	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ↔ ((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
100		ancom 466	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾) ↔ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
101	99, 100	orbi12i 928	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ∨ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾)) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
102	98, 101	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
103	97, 102	bitr4di 292	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ ((1st ‘𝑥) + 1) ∈ (1...𝑀))) ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
104	56, 103	bitrid 286	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)) ∈ (1...𝑀) ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
105		ifel 4530	. . . . . . . . . . . . . . . 16 ⊢ (if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ ((2nd ‘𝑥) + 1) ∈ (1...𝑁))))
106		simplrr 790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℕ)
107	106	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℝ)
108	16	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ ℝ)
109	108	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110		smat.n	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑁 ∈ ℕ)
111	110	nnred 12276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℝ)
112	111	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝐿)
114	107, 109, 113	ltled 11386	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝐿)
115		elfzle2 13586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (1...𝑁) → 𝐿 ≤ 𝑁)
116	15, 115	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ≤ 𝑁)
117	116	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝐿 ≤ 𝑁)
118	107, 109, 112, 114, 117	letrd 11395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝑁)
119	106, 118	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁))
120	110	nnzd 12645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		fznn 13651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℤ → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁)))
122	120, 121	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁)))
123	122	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁)))
124	119, 123	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ (1...𝑁))
125	107, 109, 112, 113, 117	ltletrd 11398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝑁)
126	110	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127		nnltlem1 12692	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
128	106, 126, 127	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
129	125, 128	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ (𝑁 − 1))
130	124, 129	2thd 268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
131	130	pm5.32da 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ∈ (1...𝑁)) ↔ ((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
132		fznn 13651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℤ → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
133	120, 132	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
134	133	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
135		simprr 785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℕ)
136	135	peano2nnd 12278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((2nd ‘𝑥) + 1) ∈ ℕ)
137	136	biantrurd 542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((2nd ‘𝑥) + 1) ≤ 𝑁 ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
138	135	nnzd 12645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℤ)
139	120	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140		zltp1le 12672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ ((2nd ‘𝑥) + 1) ≤ 𝑁))
141		zltlem1 12675	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
142	140, 141	bitr3d 284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
143	138, 139, 142	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((2nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
144	134, 137, 143	3bitr2d 310	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
145	144	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((¬ (2nd ‘𝑥) < 𝐿 ∧ ((2nd ‘𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
146	131, 145	orbi12d 932	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ ((2nd ‘𝑥) + 1) ∈ (1...𝑁))) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))))
147		pm4.42 1069	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ∨ ((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿)))
148		ancom 466	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ↔ ((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
149		ancom 466	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿) ↔ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
150	148, 149	orbi12i 928	. . . . . . . . . . . . . . . . . 18 ⊢ ((((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ∨ ((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿)) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
151	147, 150	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
152	146, 151	bitr4di 292	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ ((2nd ‘𝑥) + 1) ∈ (1...𝑁))) ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
153	105, 152	bitrid 286	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)) ∈ (1...𝑁) ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
154	104, 153	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)) ∈ (1...𝑁)) ↔ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
155	55, 154	bitrd 282	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
156	155	pm5.32da 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) → ((((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))))
157		1zzd 12653	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℤ)
158	71, 157	zsubcld 12734	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
159		fznn 13651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 − 1) ∈ ℤ → ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
160	158, 159	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
161	120, 157	zsubcld 12734	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
162		fznn 13651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℤ → ((2nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
163	161, 162	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
164	160, 163	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∧ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))))
165		an4 669	. . . . . . . . . . . . . 14 ⊢ ((((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∧ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))) ↔ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
166	164, 165	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))))
167	166	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) → (((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))))
168	156, 167	bitr4d 285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) → ((((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1)))))
169	168	pm5.32da 590	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1))))))
170		elxp6 8024	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)))
171	170	anbi1i 636	. . . . . . . . . . 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 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))))
172		anass 474	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
173	171, 172	bitri 278	. . . . . . . . . 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 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ (((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))))
174		elxp6 8024	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2nd ‘𝑥) ∈ (1...(𝑁 − 1)))))
175	169, 173, 174	3bitr4g 317	. . . . . . . . 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)))))
176	29, 34, 175	3bitrd 308	. . . . . . . 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)))))
177	176	eqrdv 2760	. . . . . . 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))))
178	25, 177	eqtrid 2809	. . . . . 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))))
179	24, 178	eqtrd 2797	. . . . 5 ⊢ (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180	179	feq2d 6690	. . . 4 ⊢ (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆 ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
181	23, 180	mpbid 235	. . 3 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
182	19	rneqd 5926	. . . . 5 ⊢ (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183		rncoss 5965	. . . . 5 ⊢ ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184	182, 183	eqsstrdi 3978	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185		frn 6714	. . . . 5 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴 ⊆ 𝐵)
186	1, 2, 185	3syl 19	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐵)
187	184, 186	sstrd 3944	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐵)
188		fss 6723	. . 3 ⊢ ((𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆 ∧ ran 𝑆 ⊆ 𝐵) → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
189	181, 187, 188	syl2anc 596	. 2 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190		reldmmap 8838	. . . . . 6 ⊢ Rel dom ↑m
191	190	ovrcl 7458	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))) → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
192	1, 191	syl 18	. . . 4 ⊢ (𝜑 → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
193	192	simpld 500	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
194		ovex 7450	. . . 4 ⊢ (1...(𝑀 − 1)) ∈ V
195		ovex 7450	. . . 4 ⊢ (1...(𝑁 − 1)) ∈ V
196	194, 195	xpex 7756	. . 3 ⊢ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197		elmapg 8842	. . 3 ⊢ ((𝐵 ∈ V ∧ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V) → (𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
198	193, 196, 197	sylancl 598	. 2 ⊢ (𝜑 → (𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199	189, 198	mpbird 260	1 ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ifcif 4485  ⟨cop 4593   class class class wbr 5107   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662   ∘ ccom 5663  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8830  ℝcr 11127  1c1 11129   + caddc 11131   < clt 11271   ≤ cle 11272   − cmin 11469  ℕcn 12261  ℤcz 12619  ...cfz 13565  subMat1csmat 34311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-smat 34312

This theorem is used by:  smatcl  34320  1smat1  34322