Theorem smatrcl 29190

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 7765	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁))) → 𝐴:((1...𝑀) × (1...𝑁))⟶𝐵)
3		ffun 5961	. . . . . . . 8 ⊢ (𝐴:((1...𝑀) × (1...𝑁))⟶𝐵 → Fun 𝐴)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → Fun 𝐴)
5		eqid 2610	. . . . . . . . 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	mpt2fun 6660	. . . . . . . 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 5842	. . . . . . 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 691	. . . . . 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 12243	. . . . . . . . . 10 ⊢ (1...𝑀) ⊆ ℕ
12		smat.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
13	11, 12	sseldi 3566	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
14		fz1ssnn 12243	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ ℕ
15		smat.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
16	14, 15	sseldi 3566	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℕ)
17		smatfval 29189	. . . . . . . . 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 1318	. . . . . . . 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	syl5eq 2656	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
20	19	funeqd 5825	. . . . . 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 246	. . . . 5 ⊢ (𝜑 → Fun 𝑆)
22		fdmrn 5977	. . . . 5 ⊢ (Fun 𝑆 ↔ 𝑆:dom 𝑆⟶ran 𝑆)
23	21, 22	sylib 207	. . . 4 ⊢ (𝜑 → 𝑆:dom 𝑆⟶ran 𝑆)
24	19	dmeqd 5248	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25		dmco 5560	. . . . . . 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 5964	. . . . . . . . . . . 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 5385	. . . . . . . . . 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 2673	. . . . . . . . 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 4859	. . . . . . . . . . . 12 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
31	5, 30	fnmpt2i 7128	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (ℕ × ℕ)
32		elpreima 6245	. . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
36	35	fveq2d 6107	. . . . . . . . . . . . . . . . . 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 6552	. . . . . . . . . . . . . . . . . 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	syl6eqr 2662	. . . . . . . . . . . . . . . . 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 4586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 < 𝐾 ↔ (1st ‘𝑥) < 𝐾))
40		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → 𝑖 = (1st ‘𝑥))
41		oveq1 6556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖 + 1) = ((1st ‘𝑥) + 1))
42	39, 40, 41	ifbieq12d 4063	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = (1st ‘𝑥) → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st ‘𝑥) < 𝐾, (1st ‘𝑥), ((1st ‘𝑥) + 1)))
43	42	opeq1d 4346	. . . . . . . . . . . . . . . . . . 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 4586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 < 𝐿 ↔ (2nd ‘𝑥) < 𝐿))
45		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → 𝑗 = (2nd ‘𝑥))
46		oveq1 6556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 + 1) = ((2nd ‘𝑥) + 1))
47	44, 45, 46	ifbieq12d 4063	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd ‘𝑥) < 𝐿, (2nd ‘𝑥), ((2nd ‘𝑥) + 1)))
48	47	opeq2d 4347	. . . . . . . . . . . . . . . . . . 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 4859	. . . . . . . . . . . . . . . . . . 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	ovmpt2 6694	. . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . 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 2672	. . . . . . . . . . . . . . 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 5070	. . . . . . . . . . . . . . 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	syl6bb 275	. . . . . . . . . . . . . 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 4079	. . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℕ)
58	57	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ∈ ℝ)
59	13	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ∈ ℝ)
60	59	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 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 10912	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	62	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℝ)
64		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝐾)
65	58, 60, 64	ltled 10064	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝐾)
66		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (1...𝑀) → 𝐾 ≤ 𝑀)
67	12, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
68	67	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 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 10073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ 𝑀)
70	57, 69	jca 553	. . . . . . . . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℤ)
72		fznn 12278	. . . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . . 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 10076	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) < 𝑀)
77	61	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → 𝑀 ∈ ℕ)
78		nnltlem1 11320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℕ ∧ 𝑀 ∈ ℕ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
79	57, 77, 78	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (1st ‘𝑥) < 𝐾) → (1st ‘𝑥) ≤ (𝑀 − 1))
81	75, 80	2thd 254	. . . . . . . . . . . . . . . . . . 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 671	. . . . . . . . . . . . . . . . . 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 12278	. . . . . . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℕ)
87	86	peano2nnd 10914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((1st ‘𝑥) + 1) ∈ ℕ)
88	87	biantrurd 528	. . . . . . . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st ‘𝑥) ∈ ℤ)
90	71	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑀 ∈ ℤ)
91		zltp1le 11304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ ((1st ‘𝑥) + 1) ≤ 𝑀))
92		zltlem1 11307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((1st ‘𝑥) < 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
93	91, 92	bitr3d 269	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑥) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((1st ‘𝑥) + 1) ≤ 𝑀 ↔ (1st ‘𝑥) ≤ (𝑀 − 1)))
94	89, 90, 93	syl2anc 691	. . . . . . . . . . . . . . . . . . . 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 295	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . 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		pm4.42 995	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ∨ ((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾)))
99		ancom 465	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ (1st ‘𝑥) < 𝐾) ↔ ((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
100		ancom 465	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) ≤ (𝑀 − 1) ∧ ¬ (1st ‘𝑥) < 𝐾) ↔ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)))
101	99, 100	orbi12i 542	. . . . . . . . . . . . . . . . . 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 263	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) ≤ (𝑀 − 1) ↔ (((1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1)) ∨ (¬ (1st ‘𝑥) < 𝐾 ∧ (1st ‘𝑥) ≤ (𝑀 − 1))))
103	97, 102	syl6bbr 277	. . . . . . . . . . . . . . . 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	syl5bb 271	. . . . . . . . . . . . . . 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 4079	. . . . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℕ)
107	106	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ∈ ℝ)
108	16	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ∈ ℝ)
109	108	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 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 10912	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℝ)
112	111	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℝ)
113		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝐿)
114	107, 109, 113	ltled 10064	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝐿)
115		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (1...𝑁) → 𝐿 ≤ 𝑁)
116	15, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐿 ≤ 𝑁)
117	116	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 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 10073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ 𝑁)
119	106, 118	jca 553	. . . . . . . . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℤ)
121		fznn 12278	. . . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . . 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 10076	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) < 𝑁)
126	110	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → 𝑁 ∈ ℕ)
127		nnltlem1 11320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
128	106, 126, 127	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (2nd ‘𝑥) < 𝐿) → (2nd ‘𝑥) ≤ (𝑁 − 1))
130	124, 129	2thd 254	. . . . . . . . . . . . . . . . . . 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 671	. . . . . . . . . . . . . . . . . 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 12278	. . . . . . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℕ)
136	135	peano2nnd 10914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → ((2nd ‘𝑥) + 1) ∈ ℕ)
137	136	biantrurd 528	. . . . . . . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd ‘𝑥) ∈ ℤ)
139	120	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)) → 𝑁 ∈ ℤ)
140		zltp1le 11304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ ((2nd ‘𝑥) + 1) ≤ 𝑁))
141		zltlem1 11307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd ‘𝑥) < 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
142	140, 141	bitr3d 269	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((2nd ‘𝑥) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd ‘𝑥) + 1) ≤ 𝑁 ↔ (2nd ‘𝑥) ≤ (𝑁 − 1)))
143	138, 139, 142	syl2anc 691	. . . . . . . . . . . . . . . . . . . 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 295	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . 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		pm4.42 995	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ∨ ((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿)))
148		ancom 465	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ (2nd ‘𝑥) < 𝐿) ↔ ((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
149		ancom 465	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑥) ≤ (𝑁 − 1) ∧ ¬ (2nd ‘𝑥) < 𝐿) ↔ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)))
150	148, 149	orbi12i 542	. . . . . . . . . . . . . . . . . 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 263	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑥) ≤ (𝑁 − 1) ↔ (((2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1)) ∨ (¬ (2nd ‘𝑥) < 𝐿 ∧ (2nd ‘𝑥) ≤ (𝑁 − 1))))
152	146, 151	syl6bbr 277	. . . . . . . . . . . . . . . 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	syl5bb 271	. . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 267	. . . . . . . . . . . . 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 671	. . . . . . . . . . . 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 11285	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℤ)
158	71, 157	zsubcld 11363	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
159		fznn 12278	. . . . . . . . . . . . . . . 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 11363	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
162		fznn 12278	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 861	. . . . . . . . . . . . . 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	syl6bb 275	. . . . . . . . . . . . 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 270	. . . . . . . . . . 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 671	. . . . . . . . . 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 7091	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ × ℕ) ↔ (𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∧ ((1st ‘𝑥) ∈ ℕ ∧ (2nd ‘𝑥) ∈ ℕ)))
171	170	anbi1i 727	. . . . . . . . . . 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 679	. . . . . . . . . . 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 263	. . . . . . . . . 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 7091	. . . . . . . . . 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 302	. . . . . . . . 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 293	. . . . . . . 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 2608	. . . . . . 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	syl5eq 2656	. . . . . 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 2644	. . . . 5 ⊢ (𝜑 → dom 𝑆 = ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))
180	179	feq2d 5944	. . . 4 ⊢ (𝜑 → (𝑆:dom 𝑆⟶ran 𝑆 ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆))
181	23, 180	mpbid 221	. . 3 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶ran 𝑆)
182	19	rneqd 5274	. . . . 5 ⊢ (𝜑 → ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183		rncoss 5307	. . . . 5 ⊢ ran (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ⊆ ran 𝐴
184	182, 183	syl6eqss 3618	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ran 𝐴)
185		frn 5966	. . . . 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 3578	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐵)
188		fss 5969	. . 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 691	. 2 ⊢ (𝜑 → 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵)
190		reldmmap 7753	. . . . . 6 ⊢ Rel dom ↑𝑚
191	190	ovrcl 6584	. . . . 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 474	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
194		ovex 6577	. . . 4 ⊢ (1...(𝑀 − 1)) ∈ V
195		ovex 6577	. . . 4 ⊢ (1...(𝑁 − 1)) ∈ V
196	194, 195	xpex 6860	. . 3 ⊢ ((1...(𝑀 − 1)) × (1...(𝑁 − 1))) ∈ V
197		elmapg 7757	. . 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 693	. 2 ⊢ (𝜑 → (𝑆 ∈ (𝐵 ↑𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))) ↔ 𝑆:((1...(𝑀 − 1)) × (1...(𝑁 − 1)))⟶𝐵))
199	189, 198	mpbird 246	1 ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ifcif 4036  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℤcz 11254  ...cfz 12197  subMat1csmat 29187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-smat 29188

This theorem is referenced by:  smatcl  29196  1smat1  29198