![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 11860 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℤ) | |
2 | zsqcl 13099 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℤ) |
4 | 1zzd 11571 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℤ) | |
5 | 3, 4 | zsubcld 11650 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℤ) |
6 | sq1 13123 | . . . . 5 ⊢ (1↑2) = 1 | |
7 | eluz2b2 11925 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
8 | 7 | simprbi 483 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
9 | 1red 10218 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
10 | eluzelre 11861 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
11 | 0le1 10714 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 1) |
13 | eluzge2nn0 11891 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ0) | |
14 | 13 | nn0ge0d 11517 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 𝐴) |
15 | 9, 10, 12, 14 | lt2sqd 13208 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < 𝐴 ↔ (1↑2) < (𝐴↑2))) |
16 | 8, 15 | mpbid 222 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1↑2) < (𝐴↑2)) |
17 | 6, 16 | syl5eqbrr 4828 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) |
18 | 10 | resqcld 13200 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
19 | 9, 18 | posdifd 10777 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < (𝐴↑2) ↔ 0 < ((𝐴↑2) − 1))) |
20 | 17, 19 | mpbid 222 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 < ((𝐴↑2) − 1)) |
21 | elnnz 11550 | . . 3 ⊢ (((𝐴↑2) − 1) ∈ ℕ ↔ (((𝐴↑2) − 1) ∈ ℤ ∧ 0 < ((𝐴↑2) − 1))) | |
22 | 5, 20, 21 | sylanbrc 701 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
23 | rmspecsqrtnq 37941 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | |
24 | 23 | eldifbd 3716 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
25 | 24 | intnand 1000 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
26 | df-squarenn 37876 | . . . . 5 ⊢ ◻NN = {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} | |
27 | 26 | eleq2i 2819 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ ◻NN ↔ ((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ}) |
28 | fveq2 6340 | . . . . . 6 ⊢ (𝑎 = ((𝐴↑2) − 1) → (√‘𝑎) = (√‘((𝐴↑2) − 1))) | |
29 | 28 | eleq1d 2812 | . . . . 5 ⊢ (𝑎 = ((𝐴↑2) − 1) → ((√‘𝑎) ∈ ℚ ↔ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
30 | 29 | elrab 3492 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} ↔ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
31 | 27, 30 | bitr2i 265 | . . 3 ⊢ ((((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ) ↔ ((𝐴↑2) − 1) ∈ ◻NN) |
32 | 25, 31 | sylnib 317 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ ((𝐴↑2) − 1) ∈ ◻NN) |
33 | 22, 32 | eldifd 3714 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 {crab 3042 ∖ cdif 3700 class class class wbr 4792 ‘cfv 6037 (class class class)co 6801 ℂcc 10097 0cc0 10099 1c1 10100 < clt 10237 ≤ cle 10238 − cmin 10429 ℕcn 11183 2c2 11233 ℤcz 11540 ℤ≥cuz 11850 ℚcq 11952 ↑cexp 13025 √csqrt 14143 ◻NNcsquarenn 37871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8501 df-inf 8502 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-n0 11456 df-z 11541 df-uz 11851 df-q 11953 df-rp 11997 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-dvds 15154 df-gcd 15390 df-numer 15616 df-denom 15617 df-squarenn 37876 |
This theorem is referenced by: rmspecfund 37945 rmxyelqirr 37946 rmxycomplete 37953 rmbaserp 37955 rmxyneg 37956 rmxm1 37970 rmxluc 37972 rmxdbl 37975 ltrmxnn0 37987 jm2.19lem1 38027 jm2.23 38034 rmxdiophlem 38053 |
Copyright terms: Public domain | W3C validator |