smatrcl - Mathbox for Thierry Arnoux

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Theorem smatrcl 32205

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 8745	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐵 ↑_m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3		ffun 6668	. . . . . . . 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 7474	. . . . . . . 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 6538	. . . . . . 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 13426	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
12		smat.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
13	11, 12	sselid 3940	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
14		fz1ssnn 13426	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
15		smat.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
16	14, 15	sselid 3940	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℕ)
17		smatfval 32204	. . . . . . . . 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 1371	. . . . . . . 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 2788	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
20	19	funeqd 6520	. . . . . 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 256	. . . . 5 ⊢ (𝜑 → Fun 𝑆)
22		fdmrn 6697	. . . . 5 ⊢ (Fun 𝑆 ↔ 𝑆:dom 𝑆⟶ran 𝑆)
23	21, 22	sylib 217	. . . 4 ⊢ (𝜑 → 𝑆:dom 𝑆⟶ran 𝑆)
24	19	dmeqd 5859	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25		dmco 6204	. . . . . . 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 6674	. . . . . . . . . . . 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 6011	. . . . . . . . . 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 2823	. . . . . . . . 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 5419	. . . . . . . . . . . 12 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
31	5, 30	fnmpoi 7994	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32		elpreima 7005	. . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
36	35	fveq2d 6843	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘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))⟩)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
37		df-ov 7354	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2^nd ‘𝑥)) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
38	36, 37	eqtr4di 2794	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ((1^st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2^nd ‘𝑥)))
39		breq1 5106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1^st ‘𝑥) → (𝑖 < 𝐾 ↔ (1^st ‘𝑥) < 𝐾))
40		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1^st ‘𝑥) → 𝑖 = (1^st ‘𝑥))
41		oveq1 7358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1^st ‘𝑥) → (𝑖 + 1) = ((1^st ‘𝑥) + 1))
42	39, 40, 41	ifbieq12d 4512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = (1^st ‘𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)))
43	42	opeq1d 4834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (1^st ‘𝑥) → ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
44		breq1 5106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2^nd ‘𝑥) → (𝑗 < 𝐿 ↔ (2^nd ‘𝑥) < 𝐿))
45		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2^nd ‘𝑥) → 𝑗 = (2^nd ‘𝑥))
46		oveq1 7358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2^nd ‘𝑥) → (𝑗 + 1) = ((2^nd ‘𝑥) + 1))
47	44, 45, 46	ifbieq12d 4512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2^nd ‘𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)))
48	47	opeq2d 4835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2^nd ‘𝑥) → ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩)
49		opex 5419	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩ ∈ V
50	43, 48, 5, 49	ovmpo 7509	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) → ((1^st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2^nd ‘𝑥)) = ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩)
51	50	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((1^st ‘𝑥)(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2^nd ‘𝑥)) = ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩)
52	38, 51	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) = ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩)
53	52	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁))))
54		opelxp 5667	. . . . . . . . . . . . . . 15 ⊢ (⟨if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)), if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1))⟩ ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)) ∈ (1...𝑁)))
55	53, 54	bitrdi 286	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ (if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)) ∈ (1...𝑁))))
56		ifel 4528	. . . . . . . . . . . . . . . 16 ⊢ (if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)) ∈ (1...𝑀) ↔ (((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ ((1^st ‘𝑥) + 1) ∈ (1...𝑀))))
57		simplrl 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ∈ ℕ)
58	57	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ∈ ℝ)
59	13	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ∈ ℝ)
60	59	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → 𝐾 ∈ ℝ)
61		smat.m	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑀 ∈ ℕ)
62	61	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) < 𝐾)
65	58, 60, 64	ltled 11261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ≤ 𝐾)
66		elfzle2 13399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1...𝑀) → 𝐾 ≤ 𝑀)
67	12, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
68	67	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → 𝐾 ≤ 𝑀)
69	58, 60, 63, 65, 68	letrd 11270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ≤ 𝑀)
70	57, 69	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ 𝑀))
71	61	nnzd 12484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℤ)
72		fznn 13463	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℤ → ((1^st ‘𝑥) ∈ (1...𝑀) ↔ ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ 𝑀)))
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1^st ‘𝑥) ∈ (1...𝑀) ↔ ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ 𝑀)))
74	73	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → ((1^st ‘𝑥) ∈ (1...𝑀) ↔ ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ 𝑀)))
75	70, 74	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ∈ (1...𝑀))
76	58, 60, 63, 64, 68	ltletrd 11273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) < 𝑀)
77	61	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78		nnltlem1 12528	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1^st ‘𝑥) < 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
79	57, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → ((1^st ‘𝑥) < 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
80	76, 79	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → (1^st ‘𝑥) ≤ (𝑀 − 1))
81	75, 80	2thd 264	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (1^st ‘𝑥) < 𝐾) → ((1^st ‘𝑥) ∈ (1...𝑀) ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
82	81	pm5.32da 579	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ∈ (1...𝑀)) ↔ ((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
83		fznn 13463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℤ → (((1^st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1^st ‘𝑥) + 1) ∈ ℕ ∧ ((1^st ‘𝑥) + 1) ≤ 𝑀)))
84	71, 83	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((1^st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1^st ‘𝑥) + 1) ∈ ℕ ∧ ((1^st ‘𝑥) + 1) ≤ 𝑀)))
85	84	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((1^st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (((1^st ‘𝑥) + 1) ∈ ℕ ∧ ((1^st ‘𝑥) + 1) ≤ 𝑀)))
86		simprl 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℕ)
87	86	peano2nnd 12128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((1^st ‘𝑥) + 1) ∈ ℕ)
88	87	biantrurd 533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((1^st ‘𝑥) + 1) ≤ 𝑀 ↔ (((1^st ‘𝑥) + 1) ∈ ℕ ∧ ((1^st ‘𝑥) + 1) ≤ 𝑀)))
89	86	nnzd 12484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
90	71	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91		zltp1le 12511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1^st ‘𝑥) < 𝑀 ↔ ((1^st ‘𝑥) + 1) ≤ 𝑀))
92		zltlem1 12514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1^st ‘𝑥) < 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
93	91, 92	bitr3d 280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1^st ‘𝑥) + 1) ≤ 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
94	89, 90, 93	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((1^st ‘𝑥) + 1) ≤ 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
95	85, 88, 94	3bitr2d 306	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((1^st ‘𝑥) + 1) ∈ (1...𝑀) ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
96	95	anbi2d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((¬ (1^st ‘𝑥) < 𝐾 ∧ ((1^st ‘𝑥) + 1) ∈ (1...𝑀)) ↔ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
97	82, 96	orbi12d 917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ ((1^st ‘𝑥) + 1) ∈ (1...𝑀))) ↔ (((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)))))
98		pm4.42 1052	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (1^st ‘𝑥) < 𝐾) ∨ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1^st ‘𝑥) < 𝐾)))
99		ancom 461	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (1^st ‘𝑥) < 𝐾) ↔ ((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)))
100		ancom 461	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1^st ‘𝑥) < 𝐾) ↔ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)))
101	99, 100	orbi12i 913	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (1^st ‘𝑥) < 𝐾) ∨ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1^st ‘𝑥) < 𝐾)) ↔ (((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
102	98, 101	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
103	97, 102	bitr4di 288	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ∈ (1...𝑀)) ∨ (¬ (1^st ‘𝑥) < 𝐾 ∧ ((1^st ‘𝑥) + 1) ∈ (1...𝑀))) ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
104	56, 103	bitrid 282	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)) ∈ (1...𝑀) ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
105		ifel 4528	. . . . . . . . . . . . . . . 16 ⊢ (if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)) ∈ (1...𝑁) ↔ (((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ ((2^nd ‘𝑥) + 1) ∈ (1...𝑁))))
106		simplrr 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ∈ ℕ)
107	106	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ∈ ℝ)
108	16	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ ℝ)
109	108	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → 𝐿 ∈ ℝ)
110		smat.n	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑁 ∈ ℕ)
111	110	nnred 12126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℝ)
112	111	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) < 𝐿)
114	107, 109, 113	ltled 11261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ≤ 𝐿)
115		elfzle2 13399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (1...𝑁) → 𝐿 ≤ 𝑁)
116	15, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ≤ 𝑁)
117	116	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → 𝐿 ≤ 𝑁)
118	107, 109, 112, 114, 117	letrd 11270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ≤ 𝑁)
119	106, 118	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ 𝑁))
120	110	nnzd 12484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		fznn 13463	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℤ → ((2^nd ‘𝑥) ∈ (1...𝑁) ↔ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ 𝑁)))
122	120, 121	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((2^nd ‘𝑥) ∈ (1...𝑁) ↔ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ 𝑁)))
123	122	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → ((2^nd ‘𝑥) ∈ (1...𝑁) ↔ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ 𝑁)))
124	119, 123	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ∈ (1...𝑁))
125	107, 109, 112, 113, 117	ltletrd 11273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) < 𝑁)
126	110	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127		nnltlem1 12528	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑥) < 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
128	106, 126, 127	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → ((2^nd ‘𝑥) < 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
129	125, 128	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → (2^nd ‘𝑥) ≤ (𝑁 − 1))
130	124, 129	2thd 264	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (2^nd ‘𝑥) < 𝐿) → ((2^nd ‘𝑥) ∈ (1...𝑁) ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
131	130	pm5.32da 579	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ∈ (1...𝑁)) ↔ ((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
132		fznn 13463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℤ → (((2^nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2^nd ‘𝑥) + 1) ∈ ℕ ∧ ((2^nd ‘𝑥) + 1) ≤ 𝑁)))
133	120, 132	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((2^nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2^nd ‘𝑥) + 1) ∈ ℕ ∧ ((2^nd ‘𝑥) + 1) ≤ 𝑁)))
134	133	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((2^nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (((2^nd ‘𝑥) + 1) ∈ ℕ ∧ ((2^nd ‘𝑥) + 1) ≤ 𝑁)))
135		simprr 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
136	135	peano2nnd 12128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((2^nd ‘𝑥) + 1) ∈ ℕ)
137	136	biantrurd 533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((2^nd ‘𝑥) + 1) ≤ 𝑁 ↔ (((2^nd ‘𝑥) + 1) ∈ ℕ ∧ ((2^nd ‘𝑥) + 1) ≤ 𝑁)))
138	135	nnzd 12484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℤ)
139	120	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140		zltp1le 12511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑥) < 𝑁 ↔ ((2^nd ‘𝑥) + 1) ≤ 𝑁))
141		zltlem1 12514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑥) < 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
142	140, 141	bitr3d 280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((2^nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2^nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
143	138, 139, 142	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((2^nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
144	134, 137, 143	3bitr2d 306	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((2^nd ‘𝑥) + 1) ∈ (1...𝑁) ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
145	144	anbi2d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((¬ (2^nd ‘𝑥) < 𝐿 ∧ ((2^nd ‘𝑥) + 1) ∈ (1...𝑁)) ↔ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
146	131, 145	orbi12d 917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ ((2^nd ‘𝑥) + 1) ∈ (1...𝑁))) ↔ (((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))))
147		pm4.42 1052	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2^nd ‘𝑥) < 𝐿) ∨ ((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2^nd ‘𝑥) < 𝐿)))
148		ancom 461	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2^nd ‘𝑥) < 𝐿) ↔ ((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
149		ancom 461	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2^nd ‘𝑥) < 𝐿) ↔ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
150	148, 149	orbi12i 913	. . . . . . . . . . . . . . . . . 18 ⊢ ((((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2^nd ‘𝑥) < 𝐿) ∨ ((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2^nd ‘𝑥) < 𝐿)) ↔ (((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
151	147, 150	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
152	146, 151	bitr4di 288	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ∈ (1...𝑁)) ∨ (¬ (2^nd ‘𝑥) < 𝐿 ∧ ((2^nd ‘𝑥) + 1) ∈ (1...𝑁))) ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
153	105, 152	bitrid 282	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)) ∈ (1...𝑁) ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
154	104, 153	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → ((if((1^st ‘𝑥) < 𝐾, (1^st ‘𝑥), ((1^st ‘𝑥) + 1)) ∈ (1...𝑀) ∧ if((2^nd ‘𝑥) < 𝐿, (2^nd ‘𝑥), ((2^nd ‘𝑥) + 1)) ∈ (1...𝑁)) ↔ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
155	55, 154	bitrd 278	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)) ↔ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
156	155	pm5.32da 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) → ((((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))))
157		1zzd 12492	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℤ)
158	71, 157	zsubcld 12570	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
159		fznn 13463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 − 1) ∈ ℤ → ((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
160	158, 159	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ↔ ((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ (𝑀 − 1))))
161	120, 157	zsubcld 12570	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
162		fznn 13463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 1) ∈ ℤ → ((2^nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
163	161, 162	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘𝑥) ∈ (1...(𝑁 − 1)) ↔ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
164	160, 163	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)) ∧ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))))
165		an4 654	. . . . . . . . . . . . . 14 ⊢ ((((1^st ‘𝑥) ∈ ℕ ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)) ∧ ((2^nd ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))) ↔ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1))))
166	164, 165	bitrdi 286	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))))
167	166	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) → (((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1))) ↔ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))))
168	156, 167	bitr4d 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) → ((((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ ((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1)))))
169	168	pm5.32da 579	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁)))) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1))))))
170		elxp6 7947	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘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 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘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 469	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘𝑥) ∈ ((1...𝑀) × (1...𝑁))) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘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 274	. . . . . . . . . 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 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ (((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘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 7947	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ (1...(𝑀 − 1)) ∧ (2^nd ‘𝑥) ∈ (1...(𝑁 − 1)))))
175	169, 173, 174	3bitr4g 313	. . . . . . . . 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 304	. . . . . . . 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 2788	. . . . . 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 2776	. . . . 5 ⊢ (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180	179	feq2d 6651	. . . 4 ⊢ (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆 ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
181	23, 180	mpbid 231	. . 3 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
182	19	rneqd 5891	. . . . 5 ⊢ (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183		rncoss 5925	. . . . 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 3996	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185		frn 6672	. . . . 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 3952	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐵)
188		fss 6682	. . 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 8732	. . . . . 6 ⊢ Rel dom ↑_m
191	190	ovrcl 7392	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
194		ovex 7384	. . . 4 ⊢ (1...(𝑀 − 1)) ∈ V
195		ovex 7384	. . . 4 ⊢ (1...(𝑁 − 1)) ∈ V
196	194, 195	xpex 7679	. . 3 ⊢ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197		elmapg 8736	. . 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 256	1 ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑_m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 ifcif 4484 ⟨cop 4590 class class class wbr 5103 × cxp 5629 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 ∘ ccom 5635 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ∈ cmpo 7353 1^st c1st 7911 2^nd c2nd 7912 ↑_m cmap 8723 ℝcr 11008 1c1 11010 + caddc 11012 < clt 11147 ≤ cle 11148 − cmin 11343 ℕcn 12111 ℤcz 12457 ...cfz 13378 subMat1csmat 32202

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-smat 32203

This theorem is referenced by: smatcl 32211 1smat1 32213

W3C validator