rmxyelqirr - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxyelqirr		Structured version Visualization version GIF version

Theorem rmxyelqirr 42880

Description: The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.)

**Assertion**
Ref	Expression
rmxyelqirr	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Distinct variable groups: 𝐴,𝑎,𝑐,𝑑 𝑁,𝑎 Allowed substitution hints: 𝑁(𝑐,𝑑)

**Proof of Theorem rmxyelqirr**
Step	Hyp	Ref	Expression
1		rmspecnonsq 42877	. . . . 5 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻_NN))
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻_NN))
3		pell14qrval 42818	. . . 4 ⊢ (((𝐴↑2) − 1) ∈ (ℕ ∖ ◻_NN) → (Pell14QR‘((𝐴↑2) − 1)) = {𝑎 ∈ ℝ ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)})
4	2, 3	syl 17	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (Pell14QR‘((𝐴↑2) − 1)) = {𝑎 ∈ ℝ ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)})
5		rabssab 4060	. . . 4 ⊢ {𝑎 ∈ ℝ ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)} ⊆ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)}
6		simpl 482	. . . . . . 7 ⊢ ((𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1) → 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
7	6	reximi 3074	. . . . . 6 ⊢ (∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1) → ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
8	7	reximi 3074	. . . . 5 ⊢ (∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1) → ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)))
9	8	ss2abi 4042	. . . 4 ⊢ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)} ⊆ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}
10	5, 9	sstri 3968	. . 3 ⊢ {𝑎 ∈ ℝ ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ (𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑)) ∧ ((𝑐↑2) − (((𝐴↑2) − 1) · (𝑑↑2))) = 1)} ⊆ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}
11	4, 10	eqsstrdi 4003	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (Pell14QR‘((𝐴↑2) − 1)) ⊆ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
12		simpr 484	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
13		rmspecfund 42879	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (PellFund‘((𝐴↑2) − 1)) = (𝐴 + (√‘((𝐴↑2) − 1))))
14	13	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (PellFund‘((𝐴↑2) − 1)) = (𝐴 + (√‘((𝐴↑2) − 1))))
15	14	eqcomd 2741	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 + (√‘((𝐴↑2) − 1))) = (PellFund‘((𝐴↑2) − 1)))
16	15	oveq1d 7418	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑁))
17		oveq2 7411	. . . . 5 ⊢ (𝑎 = 𝑁 → ((PellFund‘((𝐴↑2) − 1))↑𝑎) = ((PellFund‘((𝐴↑2) − 1))↑𝑁))
18	17	rspceeqv 3624	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑁)) → ∃𝑎 ∈ ℤ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑎))
19	12, 16, 18	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ ℤ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑎))
20		pellfund14b 42869	. . . 4 ⊢ (((𝐴↑2) − 1) ∈ (ℕ ∖ ◻_NN) → (((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ (Pell14QR‘((𝐴↑2) − 1)) ↔ ∃𝑎 ∈ ℤ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑎)))
21	2, 20	syl 17	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ (Pell14QR‘((𝐴↑2) − 1)) ↔ ∃𝑎 ∈ ℤ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑎)))
22	19, 21	mpbird 257	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ (Pell14QR‘((𝐴↑2) − 1)))
23	11, 22	sseldd 3959	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 ∃wrex 3060 {crab 3415 ∖ cdif 3923 ‘cfv 6530 (class class class)co 7403 ℝcr 11126 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11464 ℕcn 12238 2c2 12293 ℕ₀cn0 12499 ℤcz 12586 ℤ_≥cuz 12850 ↑cexp 14077 √csqrt 15250 ◻_NNcsquarenn 42806 Pell14QRcpell14qr 42809 PellFundcpellfund 42810

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-acn 9954 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-dvds 16271 df-gcd 16512 df-numer 16752 df-denom 16753 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515 df-squarenn 42811 df-pell1qr 42812 df-pell14qr 42813 df-pell1234qr 42814 df-pellfund 42815

This theorem is referenced by: rmxyelxp 42883 rmxyval 42886

W3C validator