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Mathbox for Stefan O'Rear |
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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 12887 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
2 | 1 | sqcld 14180 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℂ) |
3 | ax-1cn 11210 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subcl 11504 | . . . 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 15472 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ ℂ) |
7 | eluz2nn 12921 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
8 | 7 | nnsqcld 14279 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℕ) |
9 | nnm1nn0 12564 | . . . 4 ⊢ ((𝐴↑2) ∈ ℕ → ((𝐴↑2) − 1) ∈ ℕ0) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ0) |
11 | nnm1nn0 12564 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 − 1) ∈ ℕ0) |
13 | binom2sub1 14256 | . . . . . 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 12341 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℂ) | |
16 | 15, 1 | mulcld 11278 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℂ) |
17 | 3 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℂ) |
18 | 2, 16, 17 | subsubd 11645 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) = (((𝐴↑2) − (2 · 𝐴)) + 1)) |
19 | 14, 18 | eqtr4d 2777 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = ((𝐴↑2) − ((2 · 𝐴) − 1))) |
20 | 1red 11259 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
21 | 2re 12337 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℝ) |
23 | eluzelre 12886 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
24 | 22, 23 | remulcld 11288 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
25 | 24, 20 | resubcld 11688 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((2 · 𝐴) − 1) ∈ ℝ) |
26 | 8 | nnred 12278 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
27 | eluz2gt1 12959 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) | |
28 | 20, 20, 23, 27, 27 | lt2addmuld 12513 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 + 1) < (2 · 𝐴)) |
29 | remulcl 11237 | . . . . . . . 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 11860 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((1 + 1) < (2 · 𝐴) ↔ 1 < ((2 · 𝐴) − 1))) |
32 | 28, 31 | mpbid 232 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < ((2 · 𝐴) − 1)) |
33 | 20, 25, 26, 32 | ltsub2dd 11873 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) < ((𝐴↑2) − 1)) |
34 | 19, 33 | eqbrtrd 5169 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) < ((𝐴↑2) − 1)) |
35 | 26 | ltm1d 12197 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (𝐴↑2)) |
36 | npcan 11514 | . . . . . 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 7445 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (((𝐴 − 1) + 1)↑2) = (𝐴↑2)) |
39 | 35, 38 | breqtrrd 5175 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2)) |
40 | nonsq 16792 | . . 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 839 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
42 | 6, 41 | eldifd 3973 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 ℕcn 12263 2c2 12318 ℕ0cn0 12523 ℤ≥cuz 12875 ℚcq 12987 ↑cexp 14098 √csqrt 15268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-numer 16768 df-denom 16769 |
This theorem is referenced by: rmspecnonsq 42894 rmxypairf1o 42899 rmxycomplete 42905 rmxyneg 42908 rmxyadd 42909 rmxy1 42910 rmxy0 42911 jm2.22 42983 |
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