rmxyval - Mathbox for Stefan O'Rear

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Theorem rmxyval 40274

Description: Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)

**Assertion**
Ref	Expression
rmxyval	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 X_rm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Y_rm 𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))

Proof of Theorem rmxyval Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rmxfval 40263	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) = (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
2		rmyfval 40264	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) = (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
3	2	oveq2d 7172	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((√‘((𝐴↑2) − 1)) · (𝐴 Y_rm 𝑁)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))))
4	1, 3	oveq12d 7174	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 X_rm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Y_rm 𝑁))) = ((1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))))
5		rmxyelxp 40271	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) ∈ (ℕ₀ × ℤ))
6		fveq2 6663	. . . . 5 ⊢ (𝑎 = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) → (1^st ‘𝑎) = (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
7		fveq2 6663	. . . . . 6 ⊢ (𝑎 = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) → (2^nd ‘𝑎) = (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
8	7	oveq2d 7172	. . . . 5 ⊢ (𝑎 = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑎)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))))
9	6, 8	oveq12d 7174	. . . 4 ⊢ (𝑎 = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) → ((1^st ‘𝑎) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑎))) = ((1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))))
10		fveq2 6663	. . . . . 6 ⊢ (𝑏 = 𝑎 → (1^st ‘𝑏) = (1^st ‘𝑎))
11		fveq2 6663	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (2^nd ‘𝑏) = (2^nd ‘𝑎))
12	11	oveq2d 7172	. . . . . 6 ⊢ (𝑏 = 𝑎 → ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑎)))
13	10, 12	oveq12d 7174	. . . . 5 ⊢ (𝑏 = 𝑎 → ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑎) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑎))))
14	13	cbvmptv 5139	. . . 4 ⊢ (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))) = (𝑎 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑎) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑎))))
15		ovex 7189	. . . 4 ⊢ ((1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))) ∈ V
16	9, 14, 15	fvmpt 6764	. . 3 ⊢ ((^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) ∈ (ℕ₀ × ℤ) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) = ((1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))))
17	5, 16	syl 17	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) = ((1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))))
18		rmxypairf1o 40270	. . . 4 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
19	18	adantr 484	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
20		rmxyelqirr 40269	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
21		f1ocnvfv2 7032	. . 3 ⊢ (((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))):(ℕ₀ × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))} ∧ ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
22	19, 20, 21	syl2anc 587	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
23	4, 17, 22	3eqtr2d 2799	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 X_rm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Y_rm 𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2735 ∃wrex 3071 ↦ cmpt 5116 × cxp 5526 ^◡ccnv 5527 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7156 1^st c1st 7697 2^nd c2nd 7698 1c1 10589 + caddc 10591 · cmul 10593 − cmin 10921 2c2 11742 ℕ₀cn0 11947 ℤcz 12033 ℤ_≥cuz 12295 ↑cexp 13492 √csqrt 14653 X_rm crmx 40259 Y_rm crmy 40260

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-oadd 8122 df-omul 8123 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-acn 9417 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-xnn0 12020 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-dvds 15669 df-gcd 15907 df-numer 16144 df-denom 16145 df-struct 16557 df-ndx 16558 df-slot 16559 df-base 16561 df-sets 16562 df-ress 16563 df-plusg 16650 df-mulr 16651 df-starv 16652 df-sca 16653 df-vsca 16654 df-ip 16655 df-tset 16656 df-ple 16657 df-ds 16659 df-unif 16660 df-hom 16661 df-cco 16662 df-rest 16768 df-topn 16769 df-0g 16787 df-gsum 16788 df-topgen 16789 df-pt 16790 df-prds 16793 df-xrs 16847 df-qtop 16852 df-imas 16853 df-xps 16855 df-mre 16929 df-mrc 16930 df-acs 16932 df-mgm 17932 df-sgrp 17981 df-mnd 17992 df-submnd 18037 df-mulg 18306 df-cntz 18528 df-cmn 18989 df-psmet 20172 df-xmet 20173 df-met 20174 df-bl 20175 df-mopn 20176 df-fbas 20177 df-fg 20178 df-cnfld 20181 df-top 21608 df-topon 21625 df-topsp 21647 df-bases 21660 df-cld 21733 df-ntr 21734 df-cls 21735 df-nei 21812 df-lp 21850 df-perf 21851 df-cn 21941 df-cnp 21942 df-haus 22029 df-tx 22276 df-hmeo 22469 df-fil 22560 df-fm 22652 df-flim 22653 df-flf 22654 df-xms 23036 df-ms 23037 df-tms 23038 df-cncf 23593 df-limc 24579 df-dv 24580 df-log 25261 df-squarenn 40200 df-pell1qr 40201 df-pell14qr 40202 df-pell1234qr 40203 df-pellfund 40204 df-rmx 40261 df-rmy 40262

This theorem is referenced by: rmxycomplete 40276 rmxyneg 40279 rmxyadd 40280 rmxy1 40281 rmxy0 40282 jm2.21 40353

W3C validator