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Theorem rmyfval 39380
Description: Value of the Y sequence. Not used after rmxyval 39390 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmyfval ((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) = (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
Distinct variable groups:   𝐴,𝑏   𝑁,𝑏

Proof of Theorem rmyfval
Dummy variables 𝑛 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7152 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
21fvoveq1d 7167 . . . . . . . . 9 (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
32oveq1d 7160 . . . . . . . 8 (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))
43oveq2d 7161 . . . . . . 7 (𝑎 = 𝐴 → ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))) = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
54mpteq2dv 5153 . . . . . 6 (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
65cnveqd 5739 . . . . 5 (𝑎 = 𝐴(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
76adantr 481 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
8 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
98, 2oveq12d 7163 . . . . 5 (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10 id 22 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
119, 10oveqan12d 7164 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
127, 11fveq12d 6670 . . 3 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)) = ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))
1312fveq2d 6667 . 2 ((𝑎 = 𝐴𝑛 = 𝑁) → (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))) = (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
14 df-rmy 39378 . 2 Yrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15 fvex 6676 . 2 (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
1613, 14, 15ovmpoa 7294 1 ((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) = (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cmpt 5137   × cxp 5546  ccnv 5547  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  2c2 11680  0cn0 11885  cz 11969  cuz 12231  cexp 13417  csqrt 14580   Yrm crmy 39376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rmy 39378
This theorem is referenced by:  rmxyval  39390
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