rmxyelqirr - Mathbox for Stefan O'Rear

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Theorem rmxyelqirr 42883

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 42880	. . . . 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 42821	. . . 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 4036	. . . 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 3067	. . . . . 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 3067	. . . . 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 4019	. . . 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 3945	. . 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 3980	. 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 42882	. . . . . . 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 2735	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 + (√‘((𝐴↑2) − 1))) = (PellFund‘((𝐴↑2) − 1)))
16	15	oveq1d 7364	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) = ((PellFund‘((𝐴↑2) − 1))↑𝑁))
17		oveq2 7357	. . . . 5 ⊢ (𝑎 = 𝑁 → ((PellFund‘((𝐴↑2) − 1))↑𝑎) = ((PellFund‘((𝐴↑2) − 1))↑𝑁))
18	17	rspceeqv 3600	. . . 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 42872	. . . 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 3936	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 2109 {cab 2707 ∃wrex 3053 {crab 3394 ∖ cdif 3900 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 ℕcn 12128 2c2 12183 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ↑cexp 13968 √csqrt 15140 ◻_NNcsquarenn 42809 Pell14QRcpell14qr 42812 PellFundcpellfund 42813

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-numer 16646 df-denom 16647 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-squarenn 42814 df-pell1qr 42815 df-pell14qr 42816 df-pell1234qr 42817 df-pellfund 42818

This theorem is referenced by: rmxyelxp 42885 rmxyval 42888

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