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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmspecsqrtnq | Structured version Visualization version GIF version |
Description: The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.) |
Ref | Expression |
---|---|
rmspecsqrtnq | ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelcn 12243 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
2 | 1 | sqcld 13496 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℂ) |
3 | ax-1cn 10583 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subcl 10873 | . . . 4 ⊢ (((𝐴↑2) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑2) − 1) ∈ ℂ) | |
5 | 2, 3, 4 | sylancl 586 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℂ) |
6 | 5 | sqrtcld 14785 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ ℂ) |
7 | eluz2nn 12272 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
8 | 7 | nnsqcld 13593 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℕ) |
9 | nnm1nn0 11926 | . . . 4 ⊢ ((𝐴↑2) ∈ ℕ → ((𝐴↑2) − 1) ∈ ℕ0) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ0) |
11 | nnm1nn0 11926 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 − 1) ∈ ℕ0) |
13 | binom2sub1 13570 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1)) | |
14 | 1, 13 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1)) |
15 | 2cnd 11703 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℂ) | |
16 | 15, 1 | mulcld 10649 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℂ) |
17 | 3 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℂ) |
18 | 2, 16, 17 | subsubd 11013 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) = (((𝐴↑2) − (2 · 𝐴)) + 1)) |
19 | 14, 18 | eqtr4d 2856 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = ((𝐴↑2) − ((2 · 𝐴) − 1))) |
20 | 1red 10630 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
21 | 2re 11699 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℝ) |
23 | eluzelre 12242 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
24 | 22, 23 | remulcld 10659 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
25 | 24, 20 | resubcld 11056 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((2 · 𝐴) − 1) ∈ ℝ) |
26 | 8 | nnred 11641 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
27 | eluz2gt1 12308 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) | |
28 | 20, 20, 23, 27, 27 | lt2addmuld 11875 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 + 1) < (2 · 𝐴)) |
29 | remulcl 10610 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ) | |
30 | 21, 23, 29 | sylancr 587 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
31 | 20, 20, 30 | ltaddsubd 11228 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((1 + 1) < (2 · 𝐴) ↔ 1 < ((2 · 𝐴) − 1))) |
32 | 28, 31 | mpbid 233 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < ((2 · 𝐴) − 1)) |
33 | 20, 25, 26, 32 | ltsub2dd 11241 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) < ((𝐴↑2) − 1)) |
34 | 19, 33 | eqbrtrd 5079 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) < ((𝐴↑2) − 1)) |
35 | 26 | ltm1d 11560 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (𝐴↑2)) |
36 | npcan 10883 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
37 | 1, 3, 36 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1) + 1) = 𝐴) |
38 | 37 | oveq1d 7160 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (((𝐴 − 1) + 1)↑2) = (𝐴↑2)) |
39 | 35, 38 | breqtrrd 5085 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2)) |
40 | nonsq 16087 | . . 3 ⊢ (((((𝐴↑2) − 1) ∈ ℕ0 ∧ (𝐴 − 1) ∈ ℕ0) ∧ (((𝐴 − 1)↑2) < ((𝐴↑2) − 1) ∧ ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2))) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) | |
41 | 10, 12, 34, 39, 40 | syl22anc 834 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
42 | 6, 41 | eldifd 3944 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 − cmin 10858 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ℤ≥cuz 12231 ℚcq 12336 ↑cexp 13417 √csqrt 14580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-numer 16063 df-denom 16064 |
This theorem is referenced by: rmspecnonsq 39382 rmxypairf1o 39386 rmxycomplete 39392 rmxyneg 39395 rmxyadd 39396 rmxy1 39397 rmxy0 39398 jm2.22 39470 |
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