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

Theorem rmyfval 42892
Description: Value of the Y sequence. Not used after rmxyval 42903 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 7437 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
21fvoveq1d 7452 . . . . . . . . 9 (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
32oveq1d 7445 . . . . . . . 8 (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))
43oveq2d 7446 . . . . . . 7 (𝑎 = 𝐴 → ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))) = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
54mpteq2dv 5249 . . . . . 6 (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
65cnveqd 5888 . . . . 5 (𝑎 = 𝐴(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
76adantr 480 . . . 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 7448 . . . . 5 (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10 id 22 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
119, 10oveqan12d 7449 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
127, 11fveq12d 6913 . . 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 6910 . 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 42890 . 2 Yrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15 fvex 6919 . 2 (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
1613, 14, 15ovmpoa 7587 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 395   = wceq 1536  wcel 2105  cmpt 5230   × cxp 5686  ccnv 5687  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  1c1 11153   + caddc 11155   · cmul 11157  cmin 11489  2c2 12318  0cn0 12523  cz 12610  cuz 12875  cexp 14098  csqrt 15268   Yrm crmy 42888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rmy 42890
This theorem is referenced by:  rmxyval  42903
  Copyright terms: Public domain W3C validator