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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxdiophlem | Structured version Visualization version GIF version |
Description: X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
Ref | Expression |
---|---|
rmxdiophlem | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0sqcl 13094 | . . . . . 6 ⊢ (𝑋 ∈ ℕ0 → (𝑋↑2) ∈ ℕ0) | |
2 | 1 | 3ad2ant3 1129 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋↑2) ∈ ℕ0) |
3 | 2 | nn0cnd 11560 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋↑2) ∈ ℂ) |
4 | simp1 1130 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → 𝐴 ∈ (ℤ≥‘2)) | |
5 | nn0z 11607 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
6 | 5 | 3ad2ant2 1128 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → 𝑁 ∈ ℤ) |
7 | frmx 38004 | . . . . . . . 8 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
8 | 7 | fovcl 6916 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
9 | 4, 6, 8 | syl2anc 573 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
10 | nn0sqcl 13094 | . . . . . 6 ⊢ ((𝐴 Xrm 𝑁) ∈ ℕ0 → ((𝐴 Xrm 𝑁)↑2) ∈ ℕ0) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝐴 Xrm 𝑁)↑2) ∈ ℕ0) |
12 | 11 | nn0cnd 11560 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝐴 Xrm 𝑁)↑2) ∈ ℂ) |
13 | rmspecnonsq 37998 | . . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) | |
14 | 13 | eldifad 3735 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
15 | 14 | nnnn0d 11558 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ0) |
16 | 15 | 3ad2ant1 1127 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝐴↑2) − 1) ∈ ℕ0) |
17 | rmynn0 38050 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm 𝑁) ∈ ℕ0) | |
18 | 17 | 3adant3 1126 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝐴 Yrm 𝑁) ∈ ℕ0) |
19 | nn0sqcl 13094 | . . . . . . 7 ⊢ ((𝐴 Yrm 𝑁) ∈ ℕ0 → ((𝐴 Yrm 𝑁)↑2) ∈ ℕ0) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝐴 Yrm 𝑁)↑2) ∈ ℕ0) |
21 | 16, 20 | nn0mulcld 11563 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2)) ∈ ℕ0) |
22 | 21 | nn0cnd 11560 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2)) ∈ ℂ) |
23 | 3, 12, 22 | subcan2ad 10643 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) ↔ (𝑋↑2) = ((𝐴 Xrm 𝑁)↑2))) |
24 | rmxynorm 38009 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1) | |
25 | 4, 6, 24 | syl2anc 573 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1) |
26 | 25 | eqeq2d 2781 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) ↔ ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)) |
27 | nn0re 11508 | . . . . . 6 ⊢ (𝑋 ∈ ℕ0 → 𝑋 ∈ ℝ) | |
28 | nn0ge0 11525 | . . . . . 6 ⊢ (𝑋 ∈ ℕ0 → 0 ≤ 𝑋) | |
29 | 27, 28 | jca 501 | . . . . 5 ⊢ (𝑋 ∈ ℕ0 → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋)) |
30 | 29 | 3ad2ant3 1129 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋)) |
31 | nn0re 11508 | . . . . . 6 ⊢ ((𝐴 Xrm 𝑁) ∈ ℕ0 → (𝐴 Xrm 𝑁) ∈ ℝ) | |
32 | nn0ge0 11525 | . . . . . 6 ⊢ ((𝐴 Xrm 𝑁) ∈ ℕ0 → 0 ≤ (𝐴 Xrm 𝑁)) | |
33 | 31, 32 | jca 501 | . . . . 5 ⊢ ((𝐴 Xrm 𝑁) ∈ ℕ0 → ((𝐴 Xrm 𝑁) ∈ ℝ ∧ 0 ≤ (𝐴 Xrm 𝑁))) |
34 | 9, 33 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝐴 Xrm 𝑁) ∈ ℝ ∧ 0 ≤ (𝐴 Xrm 𝑁))) |
35 | sq11 13143 | . . . 4 ⊢ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ ((𝐴 Xrm 𝑁) ∈ ℝ ∧ 0 ≤ (𝐴 Xrm 𝑁))) → ((𝑋↑2) = ((𝐴 Xrm 𝑁)↑2) ↔ 𝑋 = (𝐴 Xrm 𝑁))) | |
36 | 30, 34, 35 | syl2anc 573 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → ((𝑋↑2) = ((𝐴 Xrm 𝑁)↑2) ↔ 𝑋 = (𝐴 Xrm 𝑁))) |
37 | 23, 26, 36 | 3bitr3rd 299 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)) |
38 | oveq1 6803 | . . . . . . 7 ⊢ (𝑦 = (𝐴 Yrm 𝑁) → (𝑦↑2) = ((𝐴 Yrm 𝑁)↑2)) | |
39 | 38 | oveq2d 6812 | . . . . . 6 ⊢ (𝑦 = (𝐴 Yrm 𝑁) → (((𝐴↑2) − 1) · (𝑦↑2)) = (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) |
40 | 39 | oveq2d 6812 | . . . . 5 ⊢ (𝑦 = (𝐴 Yrm 𝑁) → ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2)))) |
41 | 40 | eqeq1d 2773 | . . . 4 ⊢ (𝑦 = (𝐴 Yrm 𝑁) → (((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1 ↔ ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)) |
42 | 41 | ceqsrexv 3486 | . . 3 ⊢ ((𝐴 Yrm 𝑁) ∈ ℕ0 → (∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1) ↔ ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)) |
43 | 18, 42 | syl 17 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1) ↔ ((𝑋↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)) |
44 | 37, 43 | bitr4d 271 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 class class class wbr 4787 ‘cfv 6030 (class class class)co 6796 ℝcr 10141 0cc0 10142 1c1 10143 · cmul 10147 ≤ cle 10281 − cmin 10472 ℕcn 11226 2c2 11276 ℕ0cn0 11499 ℤcz 11584 ℤ≥cuz 11893 ↑cexp 13067 ◻NNcsquarenn 37926 Xrm crmx 37990 Yrm crmy 37991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-acn 8972 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-xnn0 11571 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-dvds 15190 df-gcd 15425 df-numer 15650 df-denom 15651 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851 df-log 24524 df-squarenn 37931 df-pell1qr 37932 df-pell14qr 37933 df-pell1234qr 37934 df-pellfund 37935 df-rmx 37992 df-rmy 37993 |
This theorem is referenced by: rmxdioph 38109 |
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