![]() |
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 12841 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
2 | 1 | sqcld 14116 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℂ) |
3 | ax-1cn 11174 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subcl 11466 | . . . 4 ⊢ (((𝐴↑2) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑2) − 1) ∈ ℂ) | |
5 | 2, 3, 4 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℂ) |
6 | 5 | sqrtcld 15391 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ ℂ) |
7 | eluz2nn 12875 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
8 | 7 | nnsqcld 14214 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℕ) |
9 | nnm1nn0 12520 | . . . 4 ⊢ ((𝐴↑2) ∈ ℕ → ((𝐴↑2) − 1) ∈ ℕ0) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ0) |
11 | nnm1nn0 12520 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0) | |
12 | 7, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 − 1) ∈ ℕ0) |
13 | binom2sub1 14191 | . . . . . 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 12297 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℂ) | |
16 | 15, 1 | mulcld 11241 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℂ) |
17 | 3 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℂ) |
18 | 2, 16, 17 | subsubd 11606 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) = (((𝐴↑2) − (2 · 𝐴)) + 1)) |
19 | 14, 18 | eqtr4d 2774 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = ((𝐴↑2) − ((2 · 𝐴) − 1))) |
20 | 1red 11222 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
21 | 2re 12293 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℝ) |
23 | eluzelre 12840 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
24 | 22, 23 | remulcld 11251 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
25 | 24, 20 | resubcld 11649 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((2 · 𝐴) − 1) ∈ ℝ) |
26 | 8 | nnred 12234 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) |
27 | eluz2gt1 12911 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) | |
28 | 20, 20, 23, 27, 27 | lt2addmuld 12469 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 + 1) < (2 · 𝐴)) |
29 | remulcl 11201 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ) | |
30 | 21, 23, 29 | sylancr 586 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) |
31 | 20, 20, 30 | ltaddsubd 11821 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((1 + 1) < (2 · 𝐴) ↔ 1 < ((2 · 𝐴) − 1))) |
32 | 28, 31 | mpbid 231 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < ((2 · 𝐴) − 1)) |
33 | 20, 25, 26, 32 | ltsub2dd 11834 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) < ((𝐴↑2) − 1)) |
34 | 19, 33 | eqbrtrd 5170 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) < ((𝐴↑2) − 1)) |
35 | 26 | ltm1d 12153 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (𝐴↑2)) |
36 | npcan 11476 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
37 | 1, 3, 36 | sylancl 585 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1) + 1) = 𝐴) |
38 | 37 | oveq1d 7427 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (((𝐴 − 1) + 1)↑2) = (𝐴↑2)) |
39 | 35, 38 | breqtrrd 5176 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2)) |
40 | nonsq 16702 | . . 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 836 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → ¬ (√‘((𝐴↑2) − 1)) ∈ ℚ) |
42 | 6, 41 | eldifd 3959 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 1c1 11117 + caddc 11119 · cmul 11121 < clt 11255 − cmin 11451 ℕcn 12219 2c2 12274 ℕ0cn0 12479 ℤ≥cuz 12829 ℚcq 12939 ↑cexp 14034 √csqrt 15187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-gcd 16443 df-numer 16678 df-denom 16679 |
This theorem is referenced by: rmspecnonsq 42111 rmxypairf1o 42116 rmxycomplete 42122 rmxyneg 42125 rmxyadd 42126 rmxy1 42127 rmxy0 42128 jm2.22 42200 |
Copyright terms: Public domain | W3C validator |