Theorem rmxfval 41039

Description: Value of the X sequence. Not used after rmxyval 41051 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Assertion

Ref	Expression
rmxfval	⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘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 7348	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
2	1	fvoveq1d 7363	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
3	2	oveq1d 7356	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))
4	3	oveq2d 7357	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))
5	4	mpteq2dv 5198	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))))
6	5	cnveqd 5821	. . . . 5 ⊢ (𝑎 = 𝐴 → ◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏)))) = ◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))))
7	6	adantr 482	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏)))) = ◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))))
8		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
9	8, 2	oveq12d 7359	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10		id 22	. . . . 5 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
11	9, 10	oveqan12d 7360	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
12	7, 11	fveq12d 6836	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)) = (◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))
13	12	fveq2d 6833	. 2 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))) = (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
14		df-rmx 41037	. 2 ⊢ Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15		fvex 6842	. 2 ⊢ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
16	13, 14, 15	ovmpoa 7494	1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5179   × cxp 5622  ^◡ccnv 5623  ‘cfv 6483  (class class class)co 7341  1^st c1st 7901  2^nd c2nd 7902  1c1 10977   + caddc 10979   · cmul 10981   − cmin 11310  2c2 12133  ℕ₀cn0 12338  ℤcz 12424  ℤ_≥cuz 12687  ↑cexp 13887  √csqrt 15043   X_rm crmx 41035

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3731  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6435  df-fun 6485  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-rmx 41037

This theorem is referenced by:  rmxyval  41051