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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellexlem4 | Structured version Visualization version GIF version |
Description: Lemma for pellex 42397. Invoking irrapx1 42390, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
Ref | Expression |
---|---|
pellexlem4 | ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 12251 | . . . . 5 ⊢ ℕ ∈ V | |
2 | 1, 1 | xpex 7756 | . . . 4 ⊢ (ℕ × ℕ) ∈ V |
3 | opabssxp 5770 | . . . 4 ⊢ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ⊆ (ℕ × ℕ) | |
4 | ssdomg 9021 | . . . 4 ⊢ ((ℕ × ℕ) ∈ V → ({〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ⊆ (ℕ × ℕ) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ (ℕ × ℕ))) | |
5 | 2, 3, 4 | mp2 9 | . . 3 ⊢ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ (ℕ × ℕ) |
6 | xpnnen 16191 | . . 3 ⊢ (ℕ × ℕ) ≈ ℕ | |
7 | domentr 9034 | . . 3 ⊢ (({〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ ℕ) | |
8 | 5, 6, 7 | mp2an 690 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ ℕ |
9 | nnrp 13020 | . . . . . . 7 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℝ+) | |
10 | 9 | rpsqrtcld 15394 | . . . . . 6 ⊢ (𝐷 ∈ ℕ → (√‘𝐷) ∈ ℝ+) |
11 | 10 | anim1i 613 | . . . . 5 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((√‘𝐷) ∈ ℝ+ ∧ ¬ (√‘𝐷) ∈ ℚ)) |
12 | eldif 3954 | . . . . 5 ⊢ ((√‘𝐷) ∈ (ℝ+ ∖ ℚ) ↔ ((√‘𝐷) ∈ ℝ+ ∧ ¬ (√‘𝐷) ∈ ℚ)) | |
13 | 11, 12 | sylibr 233 | . . . 4 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (√‘𝐷) ∈ (ℝ+ ∖ ℚ)) |
14 | irrapx1 42390 | . . . 4 ⊢ ((√‘𝐷) ∈ (ℝ+ ∖ ℚ) → {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))} ≈ ℕ) | |
15 | ensym 9024 | . . . 4 ⊢ ({𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))} ≈ ℕ → ℕ ≈ {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))}) | |
16 | 13, 14, 15 | 3syl 18 | . . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ℕ ≈ {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))}) |
17 | pellexlem3 42393 | . . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))} ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) | |
18 | endomtr 9033 | . . 3 ⊢ ((ℕ ≈ {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))} ∧ {𝑏 ∈ ℚ ∣ (0 < 𝑏 ∧ (abs‘(𝑏 − (√‘𝐷))) < ((denom‘𝑏)↑-2))} ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) → ℕ ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) | |
19 | 16, 17, 18 | syl2anc 582 | . 2 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ℕ ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) |
20 | sbth 9118 | . 2 ⊢ (({〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≼ ℕ ∧ ℕ ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ) | |
21 | 8, 19, 20 | sylancr 585 | 1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 {crab 3418 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 class class class wbr 5149 {copab 5211 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ≈ cen 8961 ≼ cdom 8962 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 < clt 11280 − cmin 11476 -cneg 11477 ℕcn 12245 2c2 12300 ℚcq 12965 ℝ+crp 13009 ↑cexp 14062 √csqrt 15216 abscabs 15217 denomcdenom 16709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-ico 13365 df-fz 13520 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-gcd 16473 df-numer 16710 df-denom 16711 |
This theorem is referenced by: pellexlem5 42395 |
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