rmxfval - Mathbox for Stefan O'Rear

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

Theorem rmxfval 42201

Description: Value of the X sequence. Not used after rmxyval 42213 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)

**Assertion**
Ref	Expression
rmxfval	⊢ ((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))))

Distinct variable groups: 𝐴,𝑏 𝑁,𝑏

Proof of Theorem rmxfval Dummy variables 𝑛 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7411	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
2	1	fvoveq1d 7426	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
3	2	oveq1d 7419	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))
4	3	oveq2d 7420	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))) = ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))
5	4	mpteq2dv 5243	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = (𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
6	5	cnveqd 5868	. . . . 5 ⊢ (𝑎 = 𝐴 → ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
7	6	adantr 480	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏)))) = ^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏)))))
8		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
9	8, 2	oveq12d 7422	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10		id 22	. . . . 5 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
11	9, 10	oveqan12d 7423	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
12	7, 11	fveq12d 6891	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)) = (^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))
13	12	fveq2d 6888	. 2 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (1^st ‘(^◡(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))))
14		df-rmx 42199	. 2 ⊢ X_rm = (𝑎 ∈ (ℤ_≥‘2), 𝑛 ∈ ℤ ↦ (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15		fvex 6897	. 2 ⊢ (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2^nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
16	13, 14, 15	ovmpoa 7558	1 ⊢ ((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))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5224 × cxp 5667 ^◡ccnv 5668 ‘cfv 6536 (class class class)co 7404 1^st c1st 7969 2^nd c2nd 7970 1c1 11110 + caddc 11112 · cmul 11114 − cmin 11445 2c2 12268 ℕ₀cn0 12473 ℤcz 12559 ℤ_≥cuz 12823 ↑cexp 14030 √csqrt 15184 X_rm crmx 42197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rmx 42199

This theorem is referenced by: rmxyval 42213

W3C validator