![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 13504 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℂ) |
3 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subcl 10874 | . . . 4 ⊢ (((𝐴↑2) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑2) − 1) ∈ ℂ) | |
5 | 2, 3, 4 | sylancl 589 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℂ) |
6 | 5 | sqrtcld 14789 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ ℂ) |
7 | eluz2nn 12272 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
8 | 7 | nnsqcld 13601 | . . . 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 13578 | . . . . . 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 10650 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℂ) |
17 | 3 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℂ) |
18 | 2, 16, 17 | subsubd 11014 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) = (((𝐴↑2) − (2 · 𝐴)) + 1)) |
19 | 14, 18 | eqtr4d 2836 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = ((𝐴↑2) − ((2 · 𝐴) − 1))) |
20 | 1red 10631 | . . . . 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 10660 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
25 | 24, 20 | resubcld 11057 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((2 · 𝐴) − 1) ∈ ℝ) |
26 | 8 | nnred 11640 | . . . . 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 10611 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ) | |
30 | 21, 23, 29 | sylancr 590 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
31 | 20, 20, 30 | ltaddsubd 11229 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((1 + 1) < (2 · 𝐴) ↔ 1 < ((2 · 𝐴) − 1))) |
32 | 28, 31 | mpbid 235 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < ((2 · 𝐴) − 1)) |
33 | 20, 25, 26, 32 | ltsub2dd 11242 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) < ((𝐴↑2) − 1)) |
34 | 19, 33 | eqbrtrd 5052 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) < ((𝐴↑2) − 1)) |
35 | 26 | ltm1d 11561 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (𝐴↑2)) |
36 | npcan 10884 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
37 | 1, 3, 36 | sylancl 589 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1) + 1) = 𝐴) |
38 | 37 | oveq1d 7150 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (((𝐴 − 1) + 1)↑2) = (𝐴↑2)) |
39 | 35, 38 | breqtrrd 5058 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2)) |
40 | nonsq 16089 | . . 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 837 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
42 | 6, 41 | eldifd 3892 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 − cmin 10859 ℕcn 11625 2c2 11680 ℕ0cn0 11885 ℤ≥cuz 12231 ℚcq 12336 ↑cexp 13425 √csqrt 14584 |
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 |
This theorem is referenced by: rmspecnonsq 39848 rmxypairf1o 39852 rmxycomplete 39858 rmxyneg 39861 rmxyadd 39862 rmxy1 39863 rmxy0 39864 jm2.22 39936 |
Copyright terms: Public domain | W3C validator |