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Theorem pellexlem3 41569
Description: Lemma for pellex 41573. To each good rational approximation of (βˆšβ€˜π·), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem3 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} β‰Ό {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))})
Distinct variable group:   π‘₯,𝐷,𝑦,𝑧

Proof of Theorem pellexlem3
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 12218 . . . 4 β„• ∈ V
21, 1xpex 7740 . . 3 (β„• Γ— β„•) ∈ V
3 opabssxp 5769 . . 3 {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))} βŠ† (β„• Γ— β„•)
42, 3ssexi 5323 . 2 {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))} ∈ V
5 simprl 770 . . . . . . . 8 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ π‘Ž ∈ β„š)
6 simprrl 780 . . . . . . . 8 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ 0 < π‘Ž)
7 qgt0numnn 16687 . . . . . . . 8 ((π‘Ž ∈ β„š ∧ 0 < π‘Ž) β†’ (numerβ€˜π‘Ž) ∈ β„•)
85, 6, 7syl2anc 585 . . . . . . 7 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (numerβ€˜π‘Ž) ∈ β„•)
9 qdencl 16677 . . . . . . . 8 (π‘Ž ∈ β„š β†’ (denomβ€˜π‘Ž) ∈ β„•)
105, 9syl 17 . . . . . . 7 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (denomβ€˜π‘Ž) ∈ β„•)
118, 10jca 513 . . . . . 6 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ ((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•))
12 simpll 766 . . . . . . 7 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ 𝐷 ∈ β„•)
13 simplr 768 . . . . . . 7 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ Β¬ (βˆšβ€˜π·) ∈ β„š)
14 pellexlem1 41567 . . . . . . 7 (((𝐷 ∈ β„• ∧ (numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0)
1512, 8, 10, 13, 14syl31anc 1374 . . . . . 6 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0)
16 simprrr 781 . . . . . . . 8 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2))
17 qeqnumdivden 16682 . . . . . . . . . . . 12 (π‘Ž ∈ β„š β†’ π‘Ž = ((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)))
1817oveq1d 7424 . . . . . . . . . . 11 (π‘Ž ∈ β„š β†’ (π‘Ž βˆ’ (βˆšβ€˜π·)) = (((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·)))
1918fveq2d 6896 . . . . . . . . . 10 (π‘Ž ∈ β„š β†’ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) = (absβ€˜(((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·))))
2019breq1d 5159 . . . . . . . . 9 (π‘Ž ∈ β„š β†’ ((absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2) ↔ (absβ€˜(((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))
215, 20syl 17 . . . . . . . 8 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ ((absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2) ↔ (absβ€˜(((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))
2216, 21mpbid 231 . . . . . . 7 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (absβ€˜(((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2))
23 pellexlem2 41568 . . . . . . 7 (((𝐷 ∈ β„• ∧ (numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ (absβ€˜(((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)) β†’ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))
2412, 8, 10, 22, 23syl31anc 1374 . . . . . 6 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))
2511, 15, 24jca32 517 . . . . 5 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))) β†’ (((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))))
2625ex 414 . . . 4 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ ((π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2))) β†’ (((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))))
27 breq2 5153 . . . . . 6 (π‘₯ = π‘Ž β†’ (0 < π‘₯ ↔ 0 < π‘Ž))
28 fvoveq1 7432 . . . . . . 7 (π‘₯ = π‘Ž β†’ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) = (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))))
29 fveq2 6892 . . . . . . . 8 (π‘₯ = π‘Ž β†’ (denomβ€˜π‘₯) = (denomβ€˜π‘Ž))
3029oveq1d 7424 . . . . . . 7 (π‘₯ = π‘Ž β†’ ((denomβ€˜π‘₯)↑-2) = ((denomβ€˜π‘Ž)↑-2))
3128, 30breq12d 5162 . . . . . 6 (π‘₯ = π‘Ž β†’ ((absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2) ↔ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2)))
3227, 31anbi12d 632 . . . . 5 (π‘₯ = π‘Ž β†’ ((0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2)) ↔ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2))))
3332elrab 3684 . . . 4 (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ↔ (π‘Ž ∈ β„š ∧ (0 < π‘Ž ∧ (absβ€˜(π‘Ž βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘Ž)↑-2))))
34 fvex 6905 . . . . 5 (numerβ€˜π‘Ž) ∈ V
35 fvex 6905 . . . . 5 (denomβ€˜π‘Ž) ∈ V
36 eleq1 2822 . . . . . . 7 (𝑦 = (numerβ€˜π‘Ž) β†’ (𝑦 ∈ β„• ↔ (numerβ€˜π‘Ž) ∈ β„•))
3736anbi1d 631 . . . . . 6 (𝑦 = (numerβ€˜π‘Ž) β†’ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ↔ ((numerβ€˜π‘Ž) ∈ β„• ∧ 𝑧 ∈ β„•)))
38 oveq1 7416 . . . . . . . . 9 (𝑦 = (numerβ€˜π‘Ž) β†’ (𝑦↑2) = ((numerβ€˜π‘Ž)↑2))
3938oveq1d 7424 . . . . . . . 8 (𝑦 = (numerβ€˜π‘Ž) β†’ ((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) = (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))))
4039neeq1d 3001 . . . . . . 7 (𝑦 = (numerβ€˜π‘Ž) β†’ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ↔ (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0))
4139fveq2d 6896 . . . . . . . 8 (𝑦 = (numerβ€˜π‘Ž) β†’ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) = (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))))
4241breq1d 5159 . . . . . . 7 (𝑦 = (numerβ€˜π‘Ž) β†’ ((absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))) ↔ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))
4340, 42anbi12d 632 . . . . . 6 (𝑦 = (numerβ€˜π‘Ž) β†’ ((((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))) ↔ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))))
4437, 43anbi12d 632 . . . . 5 (𝑦 = (numerβ€˜π‘Ž) β†’ (((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))) ↔ (((numerβ€˜π‘Ž) ∈ β„• ∧ 𝑧 ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))))
45 eleq1 2822 . . . . . . 7 (𝑧 = (denomβ€˜π‘Ž) β†’ (𝑧 ∈ β„• ↔ (denomβ€˜π‘Ž) ∈ β„•))
4645anbi2d 630 . . . . . 6 (𝑧 = (denomβ€˜π‘Ž) β†’ (((numerβ€˜π‘Ž) ∈ β„• ∧ 𝑧 ∈ β„•) ↔ ((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•)))
47 oveq1 7416 . . . . . . . . . 10 (𝑧 = (denomβ€˜π‘Ž) β†’ (𝑧↑2) = ((denomβ€˜π‘Ž)↑2))
4847oveq2d 7425 . . . . . . . . 9 (𝑧 = (denomβ€˜π‘Ž) β†’ (𝐷 Β· (𝑧↑2)) = (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))
4948oveq2d 7425 . . . . . . . 8 (𝑧 = (denomβ€˜π‘Ž) β†’ (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) = (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))))
5049neeq1d 3001 . . . . . . 7 (𝑧 = (denomβ€˜π‘Ž) β†’ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ↔ (((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0))
5149fveq2d 6896 . . . . . . . 8 (𝑧 = (denomβ€˜π‘Ž) β†’ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) = (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))))
5251breq1d 5159 . . . . . . 7 (𝑧 = (denomβ€˜π‘Ž) β†’ ((absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))) ↔ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))
5350, 52anbi12d 632 . . . . . 6 (𝑧 = (denomβ€˜π‘Ž) β†’ (((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))) ↔ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))))
5446, 53anbi12d 632 . . . . 5 (𝑧 = (denomβ€˜π‘Ž) β†’ ((((numerβ€˜π‘Ž) ∈ β„• ∧ 𝑧 ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))) ↔ (((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))))
5534, 35, 44, 54opelopab 5543 . . . 4 (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ ∈ {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))} ↔ (((numerβ€˜π‘Ž) ∈ β„• ∧ (denomβ€˜π‘Ž) ∈ β„•) ∧ ((((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2))) β‰  0 ∧ (absβ€˜(((numerβ€˜π‘Ž)↑2) βˆ’ (𝐷 Β· ((denomβ€˜π‘Ž)↑2)))) < (1 + (2 Β· (βˆšβ€˜π·))))))
5626, 33, 553imtr4g 296 . . 3 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} β†’ ⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ ∈ {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))}))
57 ssrab2 4078 . . . . . 6 {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} βŠ† β„š
58 simprl 770 . . . . . 6 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})) β†’ π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})
5957, 58sselid 3981 . . . . 5 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})) β†’ π‘Ž ∈ β„š)
60 simprr 772 . . . . . 6 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})) β†’ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})
6157, 60sselid 3981 . . . . 5 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})) β†’ 𝑏 ∈ β„š)
6234, 35opth 5477 . . . . . . 7 (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩ ↔ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘)))
63 simprl 770 . . . . . . . . . 10 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ (numerβ€˜π‘Ž) = (numerβ€˜π‘))
64 simprr 772 . . . . . . . . . 10 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ (denomβ€˜π‘Ž) = (denomβ€˜π‘))
6563, 64oveq12d 7427 . . . . . . . . 9 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ ((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)) = ((numerβ€˜π‘) / (denomβ€˜π‘)))
66 simpll 766 . . . . . . . . . 10 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ π‘Ž ∈ β„š)
6766, 17syl 17 . . . . . . . . 9 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ π‘Ž = ((numerβ€˜π‘Ž) / (denomβ€˜π‘Ž)))
68 simplr 768 . . . . . . . . . 10 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ 𝑏 ∈ β„š)
69 qeqnumdivden 16682 . . . . . . . . . 10 (𝑏 ∈ β„š β†’ 𝑏 = ((numerβ€˜π‘) / (denomβ€˜π‘)))
7068, 69syl 17 . . . . . . . . 9 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ 𝑏 = ((numerβ€˜π‘) / (denomβ€˜π‘)))
7165, 67, 703eqtr4d 2783 . . . . . . . 8 (((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) ∧ ((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘))) β†’ π‘Ž = 𝑏)
7271ex 414 . . . . . . 7 ((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) β†’ (((numerβ€˜π‘Ž) = (numerβ€˜π‘) ∧ (denomβ€˜π‘Ž) = (denomβ€˜π‘)) β†’ π‘Ž = 𝑏))
7362, 72biimtrid 241 . . . . . 6 ((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) β†’ (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩ β†’ π‘Ž = 𝑏))
74 fveq2 6892 . . . . . . 7 (π‘Ž = 𝑏 β†’ (numerβ€˜π‘Ž) = (numerβ€˜π‘))
75 fveq2 6892 . . . . . . 7 (π‘Ž = 𝑏 β†’ (denomβ€˜π‘Ž) = (denomβ€˜π‘))
7674, 75opeq12d 4882 . . . . . 6 (π‘Ž = 𝑏 β†’ ⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩)
7773, 76impbid1 224 . . . . 5 ((π‘Ž ∈ β„š ∧ 𝑏 ∈ β„š) β†’ (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩ ↔ π‘Ž = 𝑏))
7859, 61, 77syl2anc 585 . . . 4 (((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) ∧ (π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))})) β†’ (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩ ↔ π‘Ž = 𝑏))
7978ex 414 . . 3 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ ((π‘Ž ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} ∧ 𝑏 ∈ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))}) β†’ (⟨(numerβ€˜π‘Ž), (denomβ€˜π‘Ž)⟩ = ⟨(numerβ€˜π‘), (denomβ€˜π‘)⟩ ↔ π‘Ž = 𝑏)))
8056, 79dom2d 8989 . 2 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ ({βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))} ∈ V β†’ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} β‰Ό {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))}))
814, 80mpi 20 1 ((𝐷 ∈ β„• ∧ Β¬ (βˆšβ€˜π·) ∈ β„š) β†’ {π‘₯ ∈ β„š ∣ (0 < π‘₯ ∧ (absβ€˜(π‘₯ βˆ’ (βˆšβ€˜π·))) < ((denomβ€˜π‘₯)↑-2))} β‰Ό {βŸ¨π‘¦, π‘§βŸ© ∣ ((𝑦 ∈ β„• ∧ 𝑧 ∈ β„•) ∧ (((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2))) β‰  0 ∧ (absβ€˜((𝑦↑2) βˆ’ (𝐷 Β· (𝑧↑2)))) < (1 + (2 Β· (βˆšβ€˜π·)))))})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  {copab 5211   Γ— cxp 5675  β€˜cfv 6544  (class class class)co 7409   β‰Ό cdom 8937  0cc0 11110  1c1 11111   + caddc 11113   Β· cmul 11115   < clt 11248   βˆ’ cmin 11444  -cneg 11445   / cdiv 11871  β„•cn 12212  2c2 12267  β„šcq 12932  β†‘cexp 14027  βˆšcsqrt 15180  abscabs 15181  numercnumer 16669  denomcdenom 16670
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-dvds 16198  df-gcd 16436  df-numer 16671  df-denom 16672
This theorem is referenced by:  pellexlem4  41570
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