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Theorem smatrcl 32434
Description: Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
Hypotheses
Ref Expression
smat.s 𝑆 = (𝐾(subMat1β€˜π΄)𝐿)
smat.m (πœ‘ β†’ 𝑀 ∈ β„•)
smat.n (πœ‘ β†’ 𝑁 ∈ β„•)
smat.k (πœ‘ β†’ 𝐾 ∈ (1...𝑀))
smat.l (πœ‘ β†’ 𝐿 ∈ (1...𝑁))
smat.a (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))
Assertion
Ref Expression
smatrcl (πœ‘ β†’ 𝑆 ∈ (𝐡 ↑m ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))

Proof of Theorem smatrcl
Dummy variables 𝑖 𝑗 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smat.a . . . . . . . 8 (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))
2 elmapi 8790 . . . . . . . 8 (𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))) β†’ 𝐴:((1...𝑀) Γ— (1...𝑁))⟢𝐡)
3 ffun 6672 . . . . . . . 8 (𝐴:((1...𝑀) Γ— (1...𝑁))⟢𝐡 β†’ Fun 𝐴)
41, 2, 33syl 18 . . . . . . 7 (πœ‘ β†’ Fun 𝐴)
5 eqid 2733 . . . . . . . . 9 (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
65mpofun 7481 . . . . . . . 8 Fun (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
76a1i 11 . . . . . . 7 (πœ‘ β†’ Fun (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
8 funco 6542 . . . . . . 7 ((Fun 𝐴 ∧ Fun (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) β†’ Fun (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
94, 7, 8syl2anc 585 . . . . . 6 (πœ‘ β†’ Fun (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
10 smat.s . . . . . . . 8 𝑆 = (𝐾(subMat1β€˜π΄)𝐿)
11 fz1ssnn 13478 . . . . . . . . . 10 (1...𝑀) βŠ† β„•
12 smat.k . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ (1...𝑀))
1311, 12sselid 3943 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ β„•)
14 fz1ssnn 13478 . . . . . . . . . 10 (1...𝑁) βŠ† β„•
15 smat.l . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ (1...𝑁))
1614, 15sselid 3943 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ β„•)
17 smatfval 32433 . . . . . . . . 9 ((𝐾 ∈ β„• ∧ 𝐿 ∈ β„• ∧ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁)))) β†’ (𝐾(subMat1β€˜π΄)𝐿) = (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1813, 16, 1, 17syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (𝐾(subMat1β€˜π΄)𝐿) = (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1910, 18eqtrid 2785 . . . . . . 7 (πœ‘ β†’ 𝑆 = (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
2019funeqd 6524 . . . . . 6 (πœ‘ β†’ (Fun 𝑆 ↔ Fun (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))))
219, 20mpbird 257 . . . . 5 (πœ‘ β†’ Fun 𝑆)
22 fdmrn 6701 . . . . 5 (Fun 𝑆 ↔ 𝑆:dom π‘†βŸΆran 𝑆)
2321, 22sylib 217 . . . 4 (πœ‘ β†’ 𝑆:dom π‘†βŸΆran 𝑆)
2419dmeqd 5862 . . . . . 6 (πœ‘ β†’ dom 𝑆 = dom (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
25 dmco 6207 . . . . . . 7 dom (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) = (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ dom 𝐴)
26 fdm 6678 . . . . . . . . . . . 12 (𝐴:((1...𝑀) Γ— (1...𝑁))⟢𝐡 β†’ dom 𝐴 = ((1...𝑀) Γ— (1...𝑁)))
271, 2, 263syl 18 . . . . . . . . . . 11 (πœ‘ β†’ dom 𝐴 = ((1...𝑀) Γ— (1...𝑁)))
2827imaeq2d 6014 . . . . . . . . . 10 (πœ‘ β†’ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ dom 𝐴) = (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ ((1...𝑀) Γ— (1...𝑁))))
2928eleq2d 2820 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ dom 𝐴) ↔ π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ ((1...𝑀) Γ— (1...𝑁)))))
30 opex 5422 . . . . . . . . . . . 12 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
315, 30fnmpoi 8003 . . . . . . . . . . 11 (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (β„• Γ— β„•)
32 elpreima 7009 . . . . . . . . . . 11 ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) Fn (β„• Γ— β„•) β†’ (π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ ((1...𝑀) Γ— (1...𝑁))) ↔ (π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)))))
3331, 32ax-mp 5 . . . . . . . . . 10 (π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ ((1...𝑀) Γ— (1...𝑁))) ↔ (π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))))
3433a1i 11 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ ((1...𝑀) Γ— (1...𝑁))) ↔ (π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)))))
35 simplr 768 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
3635fveq2d 6847 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) = ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩))
37 df-ov 7361 . . . . . . . . . . . . . . . . . 18 ((1st β€˜π‘₯)(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd β€˜π‘₯)) = ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜βŸ¨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩)
3836, 37eqtr4di 2791 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) = ((1st β€˜π‘₯)(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd β€˜π‘₯)))
39 breq1 5109 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st β€˜π‘₯) β†’ (𝑖 < 𝐾 ↔ (1st β€˜π‘₯) < 𝐾))
40 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st β€˜π‘₯) β†’ 𝑖 = (1st β€˜π‘₯))
41 oveq1 7365 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (1st β€˜π‘₯) β†’ (𝑖 + 1) = ((1st β€˜π‘₯) + 1))
4239, 40, 41ifbieq12d 4515 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (1st β€˜π‘₯) β†’ if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)))
4342opeq1d 4837 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (1st β€˜π‘₯) β†’ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
44 breq1 5109 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd β€˜π‘₯) β†’ (𝑗 < 𝐿 ↔ (2nd β€˜π‘₯) < 𝐿))
45 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd β€˜π‘₯) β†’ 𝑗 = (2nd β€˜π‘₯))
46 oveq1 7365 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd β€˜π‘₯) β†’ (𝑗 + 1) = ((2nd β€˜π‘₯) + 1))
4744, 45, 46ifbieq12d 4515 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd β€˜π‘₯) β†’ if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)))
4847opeq2d 4838 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (2nd β€˜π‘₯) β†’ ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩)
49 opex 5422 . . . . . . . . . . . . . . . . . . 19 ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩ ∈ V
5043, 48, 5, 49ovmpo 7516 . . . . . . . . . . . . . . . . . 18 (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) β†’ ((1st β€˜π‘₯)(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd β€˜π‘₯)) = ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩)
5150adantl 483 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((1st β€˜π‘₯)(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)(2nd β€˜π‘₯)) = ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩)
5238, 51eqtrd 2773 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) = ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩)
5352eleq1d 2819 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)) ↔ ⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩ ∈ ((1...𝑀) Γ— (1...𝑁))))
54 opelxp 5670 . . . . . . . . . . . . . . 15 (⟨if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)), if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1))⟩ ∈ ((1...𝑀) Γ— (1...𝑁)) ↔ (if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)) ∈ (1...𝑀) ∧ if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)) ∈ (1...𝑁)))
5553, 54bitrdi 287 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)) ↔ (if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)) ∈ (1...𝑀) ∧ if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)) ∈ (1...𝑁))))
56 ifel 4531 . . . . . . . . . . . . . . . 16 (if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)) ∈ (1...𝑀) ↔ (((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ∈ (1...𝑀)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ ((1st β€˜π‘₯) + 1) ∈ (1...𝑀))))
57 simplrl 776 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ∈ β„•)
5857nnred 12173 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ∈ ℝ)
5913nnred 12173 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝐾 ∈ ℝ)
6059ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ 𝐾 ∈ ℝ)
61 smat.m . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ 𝑀 ∈ β„•)
6261nnred 12173 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝑀 ∈ ℝ)
6362ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ 𝑀 ∈ ℝ)
64 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) < 𝐾)
6558, 60, 64ltled 11308 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ≀ 𝐾)
66 elfzle2 13451 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (1...𝑀) β†’ 𝐾 ≀ 𝑀)
6712, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝐾 ≀ 𝑀)
6867ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ 𝐾 ≀ 𝑀)
6958, 60, 63, 65, 68letrd 11317 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ≀ 𝑀)
7057, 69jca 513 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ 𝑀))
7161nnzd 12531 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ 𝑀 ∈ β„€)
72 fznn 13515 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ β„€ β†’ ((1st β€˜π‘₯) ∈ (1...𝑀) ↔ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ 𝑀)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((1st β€˜π‘₯) ∈ (1...𝑀) ↔ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ 𝑀)))
7473ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ ((1st β€˜π‘₯) ∈ (1...𝑀) ↔ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ 𝑀)))
7570, 74mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ∈ (1...𝑀))
7658, 60, 63, 64, 68ltletrd 11320 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) < 𝑀)
7761ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ 𝑀 ∈ β„•)
78 nnltlem1 12575 . . . . . . . . . . . . . . . . . . . . . 22 (((1st β€˜π‘₯) ∈ β„• ∧ 𝑀 ∈ β„•) β†’ ((1st β€˜π‘₯) < 𝑀 ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
7957, 77, 78syl2anc 585 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ ((1st β€˜π‘₯) < 𝑀 ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
8076, 79mpbid 231 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))
8175, 802thd 265 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (1st β€˜π‘₯) < 𝐾) β†’ ((1st β€˜π‘₯) ∈ (1...𝑀) ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
8281pm5.32da 580 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ∈ (1...𝑀)) ↔ ((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
83 fznn 13515 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ β„€ β†’ (((1st β€˜π‘₯) + 1) ∈ (1...𝑀) ↔ (((1st β€˜π‘₯) + 1) ∈ β„• ∧ ((1st β€˜π‘₯) + 1) ≀ 𝑀)))
8471, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ (((1st β€˜π‘₯) + 1) ∈ (1...𝑀) ↔ (((1st β€˜π‘₯) + 1) ∈ β„• ∧ ((1st β€˜π‘₯) + 1) ≀ 𝑀)))
8584ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((1st β€˜π‘₯) + 1) ∈ (1...𝑀) ↔ (((1st β€˜π‘₯) + 1) ∈ β„• ∧ ((1st β€˜π‘₯) + 1) ≀ 𝑀)))
86 simprl 770 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (1st β€˜π‘₯) ∈ β„•)
8786peano2nnd 12175 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((1st β€˜π‘₯) + 1) ∈ β„•)
8887biantrurd 534 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((1st β€˜π‘₯) + 1) ≀ 𝑀 ↔ (((1st β€˜π‘₯) + 1) ∈ β„• ∧ ((1st β€˜π‘₯) + 1) ≀ 𝑀)))
8986nnzd 12531 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (1st β€˜π‘₯) ∈ β„€)
9071ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ 𝑀 ∈ β„€)
91 zltp1le 12558 . . . . . . . . . . . . . . . . . . . . . 22 (((1st β€˜π‘₯) ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ ((1st β€˜π‘₯) < 𝑀 ↔ ((1st β€˜π‘₯) + 1) ≀ 𝑀))
92 zltlem1 12561 . . . . . . . . . . . . . . . . . . . . . 22 (((1st β€˜π‘₯) ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ ((1st β€˜π‘₯) < 𝑀 ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
9391, 92bitr3d 281 . . . . . . . . . . . . . . . . . . . . 21 (((1st β€˜π‘₯) ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ (((1st β€˜π‘₯) + 1) ≀ 𝑀 ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
9489, 90, 93syl2anc 585 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((1st β€˜π‘₯) + 1) ≀ 𝑀 ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
9585, 88, 943bitr2d 307 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((1st β€˜π‘₯) + 1) ∈ (1...𝑀) ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
9695anbi2d 630 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((Β¬ (1st β€˜π‘₯) < 𝐾 ∧ ((1st β€˜π‘₯) + 1) ∈ (1...𝑀)) ↔ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
9782, 96orbi12d 918 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ∈ (1...𝑀)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ ((1st β€˜π‘₯) + 1) ∈ (1...𝑀))) ↔ (((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))))
98 pm4.42 1053 . . . . . . . . . . . . . . . . . 18 ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ↔ (((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (1st β€˜π‘₯) < 𝐾) ∨ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ Β¬ (1st β€˜π‘₯) < 𝐾)))
99 ancom 462 . . . . . . . . . . . . . . . . . . 19 (((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (1st β€˜π‘₯) < 𝐾) ↔ ((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
100 ancom 462 . . . . . . . . . . . . . . . . . . 19 (((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ Β¬ (1st β€˜π‘₯) < 𝐾) ↔ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
10199, 100orbi12i 914 . . . . . . . . . . . . . . . . . 18 ((((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (1st β€˜π‘₯) < 𝐾) ∨ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ Β¬ (1st β€˜π‘₯) < 𝐾)) ↔ (((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
10298, 101bitri 275 . . . . . . . . . . . . . . . . 17 ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ↔ (((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
10397, 102bitr4di 289 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((((1st β€˜π‘₯) < 𝐾 ∧ (1st β€˜π‘₯) ∈ (1...𝑀)) ∨ (Β¬ (1st β€˜π‘₯) < 𝐾 ∧ ((1st β€˜π‘₯) + 1) ∈ (1...𝑀))) ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
10456, 103bitrid 283 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)) ∈ (1...𝑀) ↔ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)))
105 ifel 4531 . . . . . . . . . . . . . . . 16 (if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)) ∈ (1...𝑁) ↔ (((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ∈ (1...𝑁)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ ((2nd β€˜π‘₯) + 1) ∈ (1...𝑁))))
106 simplrr 777 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ∈ β„•)
107106nnred 12173 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ∈ ℝ)
10816nnred 12173 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝐿 ∈ ℝ)
109108ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ 𝐿 ∈ ℝ)
110 smat.n . . . . . . . . . . . . . . . . . . . . . . . . 25 (πœ‘ β†’ 𝑁 ∈ β„•)
111110nnred 12173 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝑁 ∈ ℝ)
112111ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ 𝑁 ∈ ℝ)
113 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) < 𝐿)
114107, 109, 113ltled 11308 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ≀ 𝐿)
115 elfzle2 13451 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ (1...𝑁) β†’ 𝐿 ≀ 𝑁)
11615, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ 𝐿 ≀ 𝑁)
117116ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ 𝐿 ≀ 𝑁)
118107, 109, 112, 114, 117letrd 11317 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ≀ 𝑁)
119106, 118jca 513 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ 𝑁))
120110nnzd 12531 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ 𝑁 ∈ β„€)
121 fznn 13515 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„€ β†’ ((2nd β€˜π‘₯) ∈ (1...𝑁) ↔ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ 𝑁)))
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((2nd β€˜π‘₯) ∈ (1...𝑁) ↔ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ 𝑁)))
123122ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ ((2nd β€˜π‘₯) ∈ (1...𝑁) ↔ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ 𝑁)))
124119, 123mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ∈ (1...𝑁))
125107, 109, 112, 113, 117ltletrd 11320 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) < 𝑁)
126110ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ 𝑁 ∈ β„•)
127 nnltlem1 12575 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd β€˜π‘₯) ∈ β„• ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘₯) < 𝑁 ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
128106, 126, 127syl2anc 585 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ ((2nd β€˜π‘₯) < 𝑁 ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
129125, 128mpbid 231 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))
130124, 1292thd 265 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ (2nd β€˜π‘₯) < 𝐿) β†’ ((2nd β€˜π‘₯) ∈ (1...𝑁) ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
131130pm5.32da 580 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ∈ (1...𝑁)) ↔ ((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
132 fznn 13515 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„€ β†’ (((2nd β€˜π‘₯) + 1) ∈ (1...𝑁) ↔ (((2nd β€˜π‘₯) + 1) ∈ β„• ∧ ((2nd β€˜π‘₯) + 1) ≀ 𝑁)))
133120, 132syl 17 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ (((2nd β€˜π‘₯) + 1) ∈ (1...𝑁) ↔ (((2nd β€˜π‘₯) + 1) ∈ β„• ∧ ((2nd β€˜π‘₯) + 1) ≀ 𝑁)))
134133ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((2nd β€˜π‘₯) + 1) ∈ (1...𝑁) ↔ (((2nd β€˜π‘₯) + 1) ∈ β„• ∧ ((2nd β€˜π‘₯) + 1) ≀ 𝑁)))
135 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (2nd β€˜π‘₯) ∈ β„•)
136135peano2nnd 12175 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((2nd β€˜π‘₯) + 1) ∈ β„•)
137136biantrurd 534 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((2nd β€˜π‘₯) + 1) ≀ 𝑁 ↔ (((2nd β€˜π‘₯) + 1) ∈ β„• ∧ ((2nd β€˜π‘₯) + 1) ≀ 𝑁)))
138135nnzd 12531 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (2nd β€˜π‘₯) ∈ β„€)
139120ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ 𝑁 ∈ β„€)
140 zltp1le 12558 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd β€˜π‘₯) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((2nd β€˜π‘₯) < 𝑁 ↔ ((2nd β€˜π‘₯) + 1) ≀ 𝑁))
141 zltlem1 12561 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd β€˜π‘₯) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((2nd β€˜π‘₯) < 𝑁 ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
142140, 141bitr3d 281 . . . . . . . . . . . . . . . . . . . . 21 (((2nd β€˜π‘₯) ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (((2nd β€˜π‘₯) + 1) ≀ 𝑁 ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
143138, 139, 142syl2anc 585 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((2nd β€˜π‘₯) + 1) ≀ 𝑁 ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
144134, 137, 1433bitr2d 307 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((2nd β€˜π‘₯) + 1) ∈ (1...𝑁) ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
145144anbi2d 630 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ ((2nd β€˜π‘₯) + 1) ∈ (1...𝑁)) ↔ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
146131, 145orbi12d 918 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ∈ (1...𝑁)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ ((2nd β€˜π‘₯) + 1) ∈ (1...𝑁))) ↔ (((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))))
147 pm4.42 1053 . . . . . . . . . . . . . . . . . 18 ((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ↔ (((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ (2nd β€˜π‘₯) < 𝐿) ∨ ((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ Β¬ (2nd β€˜π‘₯) < 𝐿)))
148 ancom 462 . . . . . . . . . . . . . . . . . . 19 (((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ (2nd β€˜π‘₯) < 𝐿) ↔ ((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
149 ancom 462 . . . . . . . . . . . . . . . . . . 19 (((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ Β¬ (2nd β€˜π‘₯) < 𝐿) ↔ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
150148, 149orbi12i 914 . . . . . . . . . . . . . . . . . 18 ((((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ (2nd β€˜π‘₯) < 𝐿) ∨ ((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ∧ Β¬ (2nd β€˜π‘₯) < 𝐿)) ↔ (((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
151147, 150bitri 275 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1) ↔ (((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
152146, 151bitr4di 289 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((((2nd β€˜π‘₯) < 𝐿 ∧ (2nd β€˜π‘₯) ∈ (1...𝑁)) ∨ (Β¬ (2nd β€˜π‘₯) < 𝐿 ∧ ((2nd β€˜π‘₯) + 1) ∈ (1...𝑁))) ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
153105, 152bitrid 283 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)) ∈ (1...𝑁) ↔ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))
154104, 153anbi12d 632 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ ((if((1st β€˜π‘₯) < 𝐾, (1st β€˜π‘₯), ((1st β€˜π‘₯) + 1)) ∈ (1...𝑀) ∧ if((2nd β€˜π‘₯) < 𝐿, (2nd β€˜π‘₯), ((2nd β€˜π‘₯) + 1)) ∈ (1...𝑁)) ↔ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
15555, 154bitrd 279 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) β†’ (((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)) ↔ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
156155pm5.32da 580 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) β†’ ((((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))))
157 1zzd 12539 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 1 ∈ β„€)
15871, 157zsubcld 12617 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑀 βˆ’ 1) ∈ β„€)
159 fznn 13515 . . . . . . . . . . . . . . . 16 ((𝑀 βˆ’ 1) ∈ β„€ β†’ ((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ↔ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
160158, 159syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ↔ ((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1))))
161120, 157zsubcld 12617 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑁 βˆ’ 1) ∈ β„€)
162 fznn 13515 . . . . . . . . . . . . . . . 16 ((𝑁 βˆ’ 1) ∈ β„€ β†’ ((2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1)) ↔ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
163161, 162syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ((2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1)) ↔ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
164160, 163anbi12d 632 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1))) ↔ (((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)) ∧ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))))
165 an4 655 . . . . . . . . . . . . . 14 ((((1st β€˜π‘₯) ∈ β„• ∧ (1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1)) ∧ ((2nd β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))) ↔ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1))))
166164, 165bitrdi 287 . . . . . . . . . . . . 13 (πœ‘ β†’ (((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1))) ↔ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))))
167166adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) β†’ (((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1))) ↔ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((1st β€˜π‘₯) ≀ (𝑀 βˆ’ 1) ∧ (2nd β€˜π‘₯) ≀ (𝑁 βˆ’ 1)))))
168156, 167bitr4d 282 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩) β†’ ((((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ ((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1)))))
169168pm5.32da 580 . . . . . . . . . 10 (πœ‘ β†’ ((π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)))) ↔ (π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ ((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1))))))
170 elxp6 7956 . . . . . . . . . . . 12 (π‘₯ ∈ (β„• Γ— β„•) ↔ (π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)))
171170anbi1i 625 . . . . . . . . . . 11 ((π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ ((π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))))
172 anass 470 . . . . . . . . . . 11 (((π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ ((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•)) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ (π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)))))
173171, 172bitri 275 . . . . . . . . . 10 ((π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ (π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ (((1st β€˜π‘₯) ∈ β„• ∧ (2nd β€˜π‘₯) ∈ β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁)))))
174 elxp6 7956 . . . . . . . . . 10 (π‘₯ ∈ ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))) ↔ (π‘₯ = ⟨(1st β€˜π‘₯), (2nd β€˜π‘₯)⟩ ∧ ((1st β€˜π‘₯) ∈ (1...(𝑀 βˆ’ 1)) ∧ (2nd β€˜π‘₯) ∈ (1...(𝑁 βˆ’ 1)))))
175169, 173, 1743bitr4g 314 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ (β„• Γ— β„•) ∧ ((𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)β€˜π‘₯) ∈ ((1...𝑀) Γ— (1...𝑁))) ↔ π‘₯ ∈ ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))
17629, 34, 1753bitrd 305 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ dom 𝐴) ↔ π‘₯ ∈ ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))
177176eqrdv 2731 . . . . . . 7 (πœ‘ β†’ (β—‘(𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) β€œ dom 𝐴) = ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))))
17825, 177eqtrid 2785 . . . . . 6 (πœ‘ β†’ dom (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) = ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))))
17924, 178eqtrd 2773 . . . . 5 (πœ‘ β†’ dom 𝑆 = ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))))
180179feq2d 6655 . . . 4 (πœ‘ β†’ (𝑆:dom π‘†βŸΆran 𝑆 ↔ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢ran 𝑆))
18123, 180mpbid 231 . . 3 (πœ‘ β†’ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢ran 𝑆)
18219rneqd 5894 . . . . 5 (πœ‘ β†’ ran 𝑆 = ran (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
183 rncoss 5928 . . . . 5 ran (𝐴 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) βŠ† ran 𝐴
184182, 183eqsstrdi 3999 . . . 4 (πœ‘ β†’ ran 𝑆 βŠ† ran 𝐴)
185 frn 6676 . . . . 5 (𝐴:((1...𝑀) Γ— (1...𝑁))⟢𝐡 β†’ ran 𝐴 βŠ† 𝐡)
1861, 2, 1853syl 18 . . . 4 (πœ‘ β†’ ran 𝐴 βŠ† 𝐡)
187184, 186sstrd 3955 . . 3 (πœ‘ β†’ ran 𝑆 βŠ† 𝐡)
188 fss 6686 . . 3 ((𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢ran 𝑆 ∧ ran 𝑆 βŠ† 𝐡) β†’ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢𝐡)
189181, 187, 188syl2anc 585 . 2 (πœ‘ β†’ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢𝐡)
190 reldmmap 8777 . . . . . 6 Rel dom ↑m
191190ovrcl 7399 . . . . 5 (𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))) β†’ (𝐡 ∈ V ∧ ((1...𝑀) Γ— (1...𝑁)) ∈ V))
1921, 191syl 17 . . . 4 (πœ‘ β†’ (𝐡 ∈ V ∧ ((1...𝑀) Γ— (1...𝑁)) ∈ V))
193192simpld 496 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
194 ovex 7391 . . . 4 (1...(𝑀 βˆ’ 1)) ∈ V
195 ovex 7391 . . . 4 (1...(𝑁 βˆ’ 1)) ∈ V
196194, 195xpex 7688 . . 3 ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))) ∈ V
197 elmapg 8781 . . 3 ((𝐡 ∈ V ∧ ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1))) ∈ V) β†’ (𝑆 ∈ (𝐡 ↑m ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))) ↔ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢𝐡))
198193, 196, 197sylancl 587 . 2 (πœ‘ β†’ (𝑆 ∈ (𝐡 ↑m ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))) ↔ 𝑆:((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))⟢𝐡))
199189, 198mpbird 257 1 (πœ‘ β†’ 𝑆 ∈ (𝐡 ↑m ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βŠ† wss 3911  ifcif 4487  βŸ¨cop 4593   class class class wbr 5106   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β€œ cima 5637   ∘ ccom 5638  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921   ↑m cmap 8768  β„cr 11055  1c1 11057   + caddc 11059   < clt 11194   ≀ cle 11195   βˆ’ cmin 11390  β„•cn 12158  β„€cz 12504  ...cfz 13430  subMat1csmat 32431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-smat 32432
This theorem is referenced by:  smatcl  32440  1smat1  32442
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