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 12256 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℤ) | |
2 | zsqcl 13497 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℤ) |
4 | 1zzd 12016 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℤ) | |
5 | 3, 4 | zsubcld 12095 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℤ) |
6 | sq1 13561 | . . . . 5 ⊢ (1↑2) = 1 | |
7 | eluz2b2 12324 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
8 | 7 | simprbi 499 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
9 | 1red 10645 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
10 | eluzelre 12257 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
11 | 0le1 11166 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 1) |
13 | eluzge2nn0 12290 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ0) | |
14 | 13 | nn0ge0d 11961 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 𝐴) |
15 | 9, 10, 12, 14 | lt2sqd 13622 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < 𝐴 ↔ (1↑2) < (𝐴↑2))) |
16 | 8, 15 | mpbid 234 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1↑2) < (𝐴↑2)) |
17 | 6, 16 | eqbrtrrid 5105 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) |
18 | 10 | resqcld 13614 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
19 | 9, 18 | posdifd 11230 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < (𝐴↑2) ↔ 0 < ((𝐴↑2) − 1))) |
20 | 17, 19 | mpbid 234 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 < ((𝐴↑2) − 1)) |
21 | elnnz 11994 | . . 3 ⊢ (((𝐴↑2) − 1) ∈ ℕ ↔ (((𝐴↑2) − 1) ∈ ℤ ∧ 0 < ((𝐴↑2) − 1))) | |
22 | 5, 20, 21 | sylanbrc 585 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
23 | rmspecsqrtnq 39509 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | |
24 | 23 | eldifbd 3952 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
25 | 24 | intnand 491 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
26 | df-squarenn 39444 | . . . . 5 ⊢ ◻NN = {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} | |
27 | 26 | eleq2i 2907 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ ◻NN ↔ ((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ}) |
28 | fveq2 6673 | . . . . . 6 ⊢ (𝑎 = ((𝐴↑2) − 1) → (√‘𝑎) = (√‘((𝐴↑2) − 1))) | |
29 | 28 | eleq1d 2900 | . . . . 5 ⊢ (𝑎 = ((𝐴↑2) − 1) → ((√‘𝑎) ∈ ℚ ↔ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
30 | 29 | elrab 3683 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} ↔ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
31 | 27, 30 | bitr2i 278 | . . 3 ⊢ ((((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ) ↔ ((𝐴↑2) − 1) ∈ ◻NN) |
32 | 25, 31 | sylnib 330 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ ((𝐴↑2) − 1) ∈ ◻NN) |
33 | 22, 32 | eldifd 3950 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 ∖ cdif 3936 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 < clt 10678 ≤ cle 10679 − cmin 10873 ℕcn 11641 2c2 11695 ℤcz 11984 ℤ≥cuz 12246 ℚcq 12351 ↑cexp 13432 √csqrt 14595 ◻NNcsquarenn 39439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-dvds 15611 df-gcd 15847 df-numer 16078 df-denom 16079 df-squarenn 39444 |
This theorem is referenced by: rmspecfund 39512 rmxyelqirr 39513 rmxycomplete 39520 rmbaserp 39522 rmxyneg 39523 rmxm1 39537 rmxluc 39539 rmxdbl 39542 ltrmxnn0 39552 jm2.19lem1 39592 jm2.23 39599 rmxdiophlem 39618 |
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