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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 12890 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
| 2 | 1 | sqcld 14184 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℂ) | 
| 3 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 4 | subcl 11507 | . . . 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 15476 | . 2 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ ℂ) | 
| 7 | eluz2nn 12924 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
| 8 | 7 | nnsqcld 14283 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℕ) | 
| 9 | nnm1nn0 12567 | . . . 4 ⊢ ((𝐴↑2) ∈ ℕ → ((𝐴↑2) − 1) ∈ ℕ0) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ0) | 
| 11 | nnm1nn0 12567 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0) | |
| 12 | 7, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 − 1) ∈ ℕ0) | 
| 13 | binom2sub1 14260 | . . . . . 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 12344 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℂ) | |
| 16 | 15, 1 | mulcld 11281 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℂ) | 
| 17 | 3 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℂ) | 
| 18 | 2, 16, 17 | subsubd 11648 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) = (((𝐴↑2) − (2 · 𝐴)) + 1)) | 
| 19 | 14, 18 | eqtr4d 2780 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) = ((𝐴↑2) − ((2 · 𝐴) − 1))) | 
| 20 | 1red 11262 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 ∈ ℝ) | |
| 21 | 2re 12340 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
| 22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 2 ∈ ℝ) | 
| 23 | eluzelre 12889 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
| 24 | 22, 23 | remulcld 11291 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (2 · 𝐴) ∈ ℝ) | 
| 25 | 24, 20 | resubcld 11691 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((2 · 𝐴) − 1) ∈ ℝ) | 
| 26 | 8 | nnred 12281 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴↑2) ∈ ℝ) | 
| 27 | eluz2gt1 12962 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) | |
| 28 | 20, 20, 23, 27, 27 | lt2addmuld 12516 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (1 + 1) < (2 · 𝐴)) | 
| 29 | remulcl 11240 | . . . . . . . 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 11863 | . . . . . 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 11876 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − ((2 · 𝐴) − 1)) < ((𝐴↑2) − 1)) | 
| 34 | 19, 33 | eqbrtrd 5165 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 − 1)↑2) < ((𝐴↑2) − 1)) | 
| 35 | 26 | ltm1d 12200 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (𝐴↑2)) | 
| 36 | npcan 11517 | . . . . . 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 7446 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (((𝐴 − 1) + 1)↑2) = (𝐴↑2)) | 
| 39 | 35, 38 | breqtrrd 5171 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) < (((𝐴 − 1) + 1)↑2)) | 
| 40 | nonsq 16796 | . . 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 3962 | 1 ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ℤ≥cuz 12878 ℚcq 12990 ↑cexp 14102 √csqrt 15272 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-numer 16772 df-denom 16773 | 
| This theorem is referenced by: rmspecnonsq 42918 rmxypairf1o 42923 rmxycomplete 42929 rmxyneg 42932 rmxyadd 42933 rmxy1 42934 rmxy0 42935 jm2.22 43007 | 
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