Theorem rmxfval 40726

Description: Value of the X sequence. Not used after rmxyval 40737 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 7282	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
2	1	fvoveq1d 7297	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
3	2	oveq1d 7290	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))
4	3	oveq2d 7291	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))) = ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))
5	4	mpteq2dv 5176	. . . . . 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 5784	. . . . 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 481	. . . 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 7293	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10		id 22	. . . . 5 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
11	9, 10	oveqan12d 7294	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
12	7, 11	fveq12d 6781	. . 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 6778	. 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 40724	. 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 6787	. 2 ⊢ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
16	13, 14, 15	ovmpoa 7428	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 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  1c1 10872   + caddc 10874   · cmul 10876   − cmin 11205  2c2 12028  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  ↑cexp 13782  √csqrt 14944   X_rm crmx 40722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rmx 40724

This theorem is referenced by:  rmxyval  40737