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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmspecnonsq | Structured version Visualization version GIF version |
Description: The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
Ref | Expression |
---|---|
rmspecnonsq | ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12241 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℤ) | |
2 | zsqcl 13490 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℤ) |
4 | 1zzd 12001 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℤ) | |
5 | 3, 4 | zsubcld 12080 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℤ) |
6 | sq1 13554 | . . . . 5 ⊢ (1↑2) = 1 | |
7 | eluz2b2 12309 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
8 | 7 | simprbi 500 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
9 | 1red 10631 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
10 | eluzelre 12242 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
11 | 0le1 11152 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 1) |
13 | eluzge2nn0 12275 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ0) | |
14 | 13 | nn0ge0d 11946 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 𝐴) |
15 | 9, 10, 12, 14 | lt2sqd 13615 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < 𝐴 ↔ (1↑2) < (𝐴↑2))) |
16 | 8, 15 | mpbid 235 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1↑2) < (𝐴↑2)) |
17 | 6, 16 | eqbrtrrid 5066 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) |
18 | 10 | resqcld 13607 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
19 | 9, 18 | posdifd 11216 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < (𝐴↑2) ↔ 0 < ((𝐴↑2) − 1))) |
20 | 17, 19 | mpbid 235 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 < ((𝐴↑2) − 1)) |
21 | elnnz 11979 | . . 3 ⊢ (((𝐴↑2) − 1) ∈ ℕ ↔ (((𝐴↑2) − 1) ∈ ℤ ∧ 0 < ((𝐴↑2) − 1))) | |
22 | 5, 20, 21 | sylanbrc 586 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
23 | rmspecsqrtnq 39847 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | |
24 | 23 | eldifbd 3894 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
25 | 24 | intnand 492 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
26 | df-squarenn 39782 | . . . . 5 ⊢ ◻NN = {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} | |
27 | 26 | eleq2i 2881 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ ◻NN ↔ ((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ}) |
28 | fveq2 6645 | . . . . . 6 ⊢ (𝑎 = ((𝐴↑2) − 1) → (√‘𝑎) = (√‘((𝐴↑2) − 1))) | |
29 | 28 | eleq1d 2874 | . . . . 5 ⊢ (𝑎 = ((𝐴↑2) − 1) → ((√‘𝑎) ∈ ℚ ↔ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
30 | 29 | elrab 3628 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} ↔ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
31 | 27, 30 | bitr2i 279 | . . 3 ⊢ ((((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ) ↔ ((𝐴↑2) − 1) ∈ ◻NN) |
32 | 25, 31 | sylnib 331 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ ((𝐴↑2) − 1) ∈ ◻NN) |
33 | 22, 32 | eldifd 3892 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ∖ cdif 3878 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 2c2 11680 ℤcz 11969 ℤ≥cuz 12231 ℚcq 12336 ↑cexp 13425 √csqrt 14584 ◻NNcsquarenn 39777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-numer 16065 df-denom 16066 df-squarenn 39782 |
This theorem is referenced by: rmspecfund 39850 rmxyelqirr 39851 rmxycomplete 39858 rmbaserp 39860 rmxyneg 39861 rmxm1 39875 rmxluc 39877 rmxdbl 39880 ltrmxnn0 39890 jm2.19lem1 39930 jm2.23 39937 rmxdiophlem 39956 |
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