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 11573 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℤ) | |
2 | zsqcl 12796 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℤ) |
4 | 1zzd 11285 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℤ) | |
5 | 3, 4 | zsubcld 11363 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℤ) |
6 | sq1 12820 | . . . . 5 ⊢ (1↑2) = 1 | |
7 | eluz2b2 11637 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
8 | 7 | simprbi 479 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
9 | 1red 9934 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
10 | eluzelre 11574 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
11 | 0le1 10430 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 1) |
13 | eluzge2nn0 11603 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ0) | |
14 | 13 | nn0ge0d 11231 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ 𝐴) |
15 | 9, 10, 12, 14 | lt2sqd 12905 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < 𝐴 ↔ (1↑2) < (𝐴↑2))) |
16 | 8, 15 | mpbid 221 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1↑2) < (𝐴↑2)) |
17 | 6, 16 | syl5eqbrr 4619 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < (𝐴↑2)) |
18 | 10 | resqcld 12897 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
19 | 9, 18 | posdifd 10493 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 < (𝐴↑2) ↔ 0 < ((𝐴↑2) − 1))) |
20 | 17, 19 | mpbid 221 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 < ((𝐴↑2) − 1)) |
21 | elnnz 11264 | . . 3 ⊢ (((𝐴↑2) − 1) ∈ ℕ ↔ (((𝐴↑2) − 1) ∈ ℤ ∧ 0 < ((𝐴↑2) − 1))) | |
22 | 5, 20, 21 | sylanbrc 695 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
23 | rmspecsqrtnq 36488 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | |
24 | 23 | eldifbd 3553 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
25 | 24 | intnand 953 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
26 | df-squarenn 36423 | . . . . 5 ⊢ ◻NN = {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} | |
27 | 26 | eleq2i 2680 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ ◻NN ↔ ((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ}) |
28 | fveq2 6103 | . . . . . 6 ⊢ (𝑎 = ((𝐴↑2) − 1) → (√‘𝑎) = (√‘((𝐴↑2) − 1))) | |
29 | 28 | eleq1d 2672 | . . . . 5 ⊢ (𝑎 = ((𝐴↑2) − 1) → ((√‘𝑎) ∈ ℚ ↔ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
30 | 29 | elrab 3331 | . . . 4 ⊢ (((𝐴↑2) − 1) ∈ {𝑎 ∈ ℕ ∣ (√‘𝑎) ∈ ℚ} ↔ (((𝐴↑2) − 1) ∈ ℕ ∧ (√‘((𝐴↑2) − 1)) ∈ ℚ)) |
31 | 27, 30 | bitr2i 264 | . . 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 3551 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℤcz 11254 ℤ≥cuz 11563 ℚcq 11664 ↑cexp 12722 √csqrt 13821 ◻NNcsquarenn 36418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-numer 15281 df-denom 15282 df-squarenn 36423 |
This theorem is referenced by: rmspecfund 36492 rmxyelqirr 36493 rmxycomplete 36500 rmbaserp 36502 rmxyneg 36503 rmxm1 36517 rmxluc 36519 rmxdbl 36522 ltrmxnn0 36534 jm2.19lem1 36574 jm2.23 36581 rmxdiophlem 36600 |
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