Theorem smatrcl 33953

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 8786	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3		ffun 6665	. . . . . . . 8 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → Fun 𝐴)
5		eqid 2736	. . . . . . . . 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 7482	. . . . . . . 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 6532	. . . . . . 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 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 13471	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
12		smat.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
13	11, 12	sselid 3931	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
14		fz1ssnn 13471	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
15		smat.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
16	14, 15	sselid 3931	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℕ)
17		smatfval 33952	. . . . . . . . 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 1373	. . . . . . . 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 2783	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
20	19	funeqd 6514	. . . . . 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 257	. . . . 5 ⊢ (𝜑 → Fun 𝑆)
22		fdmrn 6693	. . . . 5 ⊢ (Fun 𝑆 ↔ 𝑆:dom 𝑆⟶ran 𝑆)
23	21, 22	sylib 218	. . . 4 ⊢ (𝜑 → 𝑆:dom 𝑆⟶ran 𝑆)
24	19	dmeqd 5854	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25		dmco 6213	. . . . . . 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 6671	. . . . . . . . . . . 12 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
27	1, 2, 26	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐴 = ((1...𝑀) × (1...𝑁)))
28	27	imaeq2d 6019	. . . . . . . . . 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 2822	. . . . . . . . 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 5412	. . . . . . . . . . . 12 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
31	5, 30	fnmpoi 8014	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32		elpreima 7003	. . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
36	35	fveq2d 6838	. . . . . . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 < 𝐾 ↔ (1st ‘𝑥) < 𝐾))
40		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → 𝑖 = (1st ‘𝑥))
41		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 + 1) = ((1st ‘𝑥) + 1))
42	39, 40, 41	ifbieq12d 4508	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = (1st ‘𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)))
43	42	opeq1d 4835	. . . . . . . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 < 𝐿 ↔ (2nd ‘𝑥) < 𝐿))
45		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → 𝑗 = (2nd ‘𝑥))
46		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 + 1) = ((2nd ‘𝑥) + 1))
47	44, 45, 46	ifbieq12d 4508	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)))
48	47	opeq2d 4836	. . . . . . . . . . . . . . . . . . 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 5412	. . . . . . . . . . . . . . . . . . 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 7518	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 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 5660	. . . . . . . . . . . . . . 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 287	. . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℕ)
58	57	nnred 12160	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℝ)
59	13	nnred 12160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ∈ ℝ)
60	59	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 12160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝐾)
65	58, 60, 64	ltled 11281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝐾)
66		elfzle2 13444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1...𝑀) → 𝐾 ≤ 𝑀)
67	12, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
68	67	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝑀)
70	57, 69	jca 511	. . . . . . . . . . . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℤ)
72		fznn 13508	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℤ → ((1st ‘𝑥) ∈ (1...𝑀) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀)))
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1st ‘𝑥) ∈ (1...𝑀) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ 𝑀)))
74	73	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . 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 11293	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝑀)
77	61	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78		nnltlem1 12559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
79	57, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ (𝑀 − 1))
81	75, 80	2thd 265	. . . . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . . . . 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 13508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℤ → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
84	71, 83	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((1st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1st ‘𝑥) + 1) ∈ ℕ ∧ ((1st ‘𝑥) + 1) ≤ 𝑀)))
85	84	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℕ)
87	86	peano2nnd 12162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((1st ‘𝑥) + 1) ∈ ℕ)
88	87	biantrurd 532	. . . . . . . . . . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℤ)
90	71	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91		zltp1le 12541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ ((1st ‘𝑥) + 1) ≤ 𝑀))
92		zltlem1 12544	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
93	91, 92	bitr3d 281	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st ‘𝑥) + 1) ≤ 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
94	89, 90, 93	syl2anc 584	. . . . . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . 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 918	. . . . . . . . . . . . . . . . 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 1053	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ∨ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾)))
99		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ↔ ((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
100		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾) ↔ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
101	99, 100	orbi12i 914	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
103	97, 102	bitr4di 289	. . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℕ)
107	106	nnred 12160	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℝ)
108	16	nnred 12160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ ℝ)
109	108	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 12160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℝ)
112	111	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝐿)
114	107, 109, 113	ltled 11281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝐿)
115		elfzle2 13444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (1...𝑁) → 𝐿 ≤ 𝑁)
116	15, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ≤ 𝑁)
117	116	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝑁)
119	106, 118	jca 511	. . . . . . . . . . . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		fznn 13508	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℤ → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁)))
122	120, 121	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ (1...𝑁) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ 𝑁)))
123	122	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . 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 11293	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝑁)
126	110	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127		nnltlem1 12559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
128	106, 126, 127	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ (𝑁 − 1))
130	124, 129	2thd 265	. . . . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . . . . 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 13508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℤ → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
133	120, 132	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((2nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2nd ‘𝑥) + 1) ∈ ℕ ∧ ((2nd ‘𝑥) + 1) ≤ 𝑁)))
134	133	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℕ)
136	135	peano2nnd 12162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((2nd ‘𝑥) + 1) ∈ ℕ)
137	136	biantrurd 532	. . . . . . . . . . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℤ)
139	120	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140		zltp1le 12541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ ((2nd ‘𝑥) + 1) ≤ 𝑁))
141		zltlem1 12544	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
142	140, 141	bitr3d 281	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
143	138, 139, 142	syl2anc 584	. . . . . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . 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 918	. . . . . . . . . . . . . . . . 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 1053	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ∨ ((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿)))
148		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ↔ ((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
149		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿) ↔ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
150	148, 149	orbi12i 914	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
152	146, 151	bitr4di 289	. . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 279	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 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 12522	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℤ)
158	71, 157	zsubcld 12601	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
159		fznn 13508	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 − 1) ∈ ℤ → ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
160	158, 159	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1st ‘𝑥) ∈ ℕ ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
161	120, 157	zsubcld 12601	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
162		fznn 13508	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℤ → ((2nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
163	161, 162	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2nd ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
164	160, 163	anbi12d 632	. . . . . . . . . . . . . 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 656	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 579	. . . . . . . . . 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 7967	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)))
171	170	anbi1i 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 = ⟨(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 468	. . . . . . . . . . 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 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 = ⟨(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 7967	. . . . . . . . . 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 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)))))
176	29, 34, 175	3bitrd 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)))))
177	176	eqrdv 2734	. . . . . . 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 2783	. . . . . 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 2771	. . . . 5 ⊢ (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180	179	feq2d 6646	. . . 4 ⊢ (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆 ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
181	23, 180	mpbid 232	. . 3 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
182	19	rneqd 5887	. . . . 5 ⊢ (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183		rncoss 5926	. . . . 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 6669	. . . . 5 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → ran 𝐴 ⊆ 𝐵)
186	1, 2, 185	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ 𝐵)
187	184, 186	sstrd 3944	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐵)
188		fss 6678	. . 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 584	. 2 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190		reldmmap 8772	. . . . . 6 ⊢ Rel dom ↑m
191	190	ovrcl 7399	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))) → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
192	1, 191	syl 17	. . . 4 ⊢ (𝜑 → (𝐵 ∈ V ∧ ((1...𝑀) × (1...𝑁)) ∈ V))
193	192	simpld 494	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
194		ovex 7391	. . . 4 ⊢ (1...(𝑀 − 1)) ∈ V
195		ovex 7391	. . . 4 ⊢ (1...(𝑁 − 1)) ∈ V
196	194, 195	xpex 7698	. . 3 ⊢ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197		elmapg 8776	. . 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 586	. 2 ⊢ (𝜑 → (𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199	189, 198	mpbird 257	1 ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  ifcif 4479  ⟨cop 4586   class class class wbr 5098   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627   ∘ ccom 5628  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8763  ℝcr 11025  1c1 11027   + caddc 11029   < clt 11166   ≤ cle 11167   − cmin 11364  ℕcn 12145  ℤcz 12488  ...cfz 13423  subMat1csmat 33950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-smat 33951

This theorem is referenced by:  smatcl  33959  1smat1  33961