smatrcl - Mathbox for Thierry Arnoux

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Theorem smatrcl 33765

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 8783	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐵 ↑_m ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3		ffun 6659	. . . . . . . 8 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → Fun 𝐴)
5		eqid 2729	. . . . . . . . 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 7477	. . . . . . . 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 6526	. . . . . . 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 13476	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
12		smat.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
13	11, 12	sselid 3935	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
14		fz1ssnn 13476	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
15		smat.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
16	14, 15	sselid 3935	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℕ)
17		smatfval 33764	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
20	19	funeqd 6508	. . . . . 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 6687	. . . . 5 ⊢ (Fun 𝑆 ↔ 𝑆:dom 𝑆⟶ran 𝑆)
23	21, 22	sylib 218	. . . 4 ⊢ (𝜑 → 𝑆:dom 𝑆⟶ran 𝑆)
24	19	dmeqd 5852	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25		dmco 6207	. . . . . . 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 6665	. . . . . . . . . . . 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 6015	. . . . . . . . . 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 2814	. . . . . . . . 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 5411	. . . . . . . . . . . 12 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
31	5, 30	fnmpoi 8012	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32		elpreima 6996	. . . . . . . . . . 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 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
36	35	fveq2d 6830	. . . . . . . . . . . . . . . . . 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 7356	. . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . . . . . . . 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 7360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1^st ‘𝑥) → (𝑖 + 1) = ((1^st ‘𝑥) + 1))
42	39, 40, 41	ifbieq12d 4507	. . . . . . . . . . . . . . . . . . . 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 4833	. . . . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . . . . . . . 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 7360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2^nd ‘𝑥) → (𝑗 + 1) = ((2^nd ‘𝑥) + 1))
47	44, 45, 46	ifbieq12d 4507	. . . . . . . . . . . . . . . . . . . 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 4834	. . . . . . . . . . . . . . . . . . 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 5411	. . . . . . . . . . . . . . . . . . 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 7513	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . 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 5659	. . . . . . . . . . . . . . 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 287	. . . . . . . . . . . . . 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 4523	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ∈ ℝ)
60	59	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . . . . . . . . . . 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 13449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1...𝑀) → 𝐾 ≤ 𝑀)
67	12, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
68	67	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℤ)
72		fznn 13513	. . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . 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 11294	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 12561	. . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 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 265	. . . . . . . . . . . . . . . . . . 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 13513	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℕ)
87	86	peano2nnd 12163	. . . . . . . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
90	71	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91		zltp1le 12543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1^st ‘𝑥) < 𝑀 ↔ ((1^st ‘𝑥) + 1) ≤ 𝑀))
92		zltlem1 12546	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1^st ‘𝑥) < 𝑀 ↔ (1^st ‘𝑥) ≤ (𝑀 − 1)))
93	91, 92	bitr3d 281	. . . . . . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . 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 918	. . . . . . . . . . . . . . . . 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 1053	. . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ (1^st ‘𝑥) < 𝐾) ↔ ((1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)))
100		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1^st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1^st ‘𝑥) < 𝐾) ↔ (¬ (1^st ‘𝑥) < 𝐾 ∧ (1^st ‘𝑥) ≤ (𝑀 − 1)))
101	99, 100	orbi12i 914	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . 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 4523	. . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ ℝ)
109	108	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℝ)
112	111	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . . . . . . . . . . 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 13449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (1...𝑁) → 𝐿 ≤ 𝑁)
116	15, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ≤ 𝑁)
117	116	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		fznn 13513	. . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . 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 11294	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 12561	. . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 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 265	. . . . . . . . . . . . . . . . . . 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 13513	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
136	135	peano2nnd 12163	. . . . . . . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℤ)
139	120	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) ∧ ((1^st ‘𝑥) ∈ ℕ ∧ (2^nd ‘𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140		zltp1le 12543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑥) < 𝑁 ↔ ((2^nd ‘𝑥) + 1) ≤ 𝑁))
141		zltlem1 12546	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^nd ‘𝑥) < 𝑁 ↔ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
142	140, 141	bitr3d 281	. . . . . . . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . 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 918	. . . . . . . . . . . . . . . . 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 1053	. . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2^nd ‘𝑥) < 𝐿) ↔ ((2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
149		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2^nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2^nd ‘𝑥) < 𝐿) ↔ (¬ (2^nd ‘𝑥) < 𝐿 ∧ (2^nd ‘𝑥) ≤ (𝑁 − 1)))
150	148, 149	orbi12i 914	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 279	. . . . . . . . . . . . 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 12524	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℤ)
158	71, 157	zsubcld 12603	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
159		fznn 13513	. . . . . . . . . . . . . . . 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 12603	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
162		fznn 13513	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 656	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 7965	. . . . . . . . . . . 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 468	. . . . . . . . . . 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 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 = ⟨(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 7965	. . . . . . . . . 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 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 2727	. . . . . . 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 2776	. . . . . 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 2764	. . . . 5 ⊢ (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180	179	feq2d 6640	. . . 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 5884	. . . . 5 ⊢ (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183		rncoss 5922	. . . . 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 3982	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185		frn 6663	. . . . 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 3948	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐵)
188		fss 6672	. . 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 8769	. . . . . 6 ⊢ Rel dom ↑_m
191	190	ovrcl 7394	. . . . 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 7386	. . . 4 ⊢ (1...(𝑀 − 1)) ∈ V
195		ovex 7386	. . . 4 ⊢ (1...(𝑁 − 1)) ∈ V
196	194, 195	xpex 7693	. . 3 ⊢ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197		elmapg 8773	. . 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)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ⊆ wss 3905 ifcif 4478 ⟨cop 4585 class class class wbr 5095 × cxp 5621 ^◡ccnv 5622 dom cdm 5623 ran crn 5624 “ cima 5626 ∘ ccom 5627 Fun wfun 6480 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8760 ℝcr 11027 1c1 11029 + caddc 11031 < clt 11168 ≤ cle 11169 − cmin 11365 ℕcn 12146 ℤcz 12489 ...cfz 13428 subMat1csmat 33762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-smat 33763

This theorem is referenced by: smatcl 33771 1smat1 33773

W3C validator