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Theorem rmyfval 36988
Description: Value of the Y sequence. Not used after rmxyval 36999 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 6622 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
21oveq1d 6630 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑎↑2) − 1) = ((𝐴↑2) − 1))
32fveq2d 6162 . . . . . . . . 9 (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
43oveq1d 6630 . . . . . . . 8 (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))
54oveq2d 6631 . . . . . . 7 (𝑎 = 𝐴 → ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))) = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
65mpteq2dv 4715 . . . . . 6 (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
76cnveqd 5268 . . . . 5 (𝑎 = 𝐴(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
87adantr 481 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
9 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
109, 3oveq12d 6633 . . . . 5 (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
11 id 22 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
1210, 11oveqan12d 6634 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
138, 12fveq12d 6164 . . 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)))
1413fveq2d 6162 . 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))))
15 df-rmy 36986 . 2 Yrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
16 fvex 6168 . 2 (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
1714, 15, 16ovmpt2a 6756 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 384   = wceq 1480  wcel 1987  cmpt 4683   × cxp 5082  ccnv 5083  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  1c1 9897   + caddc 9899   · cmul 9901  cmin 10226  2c2 11030  0cn0 11252  cz 11337  cuz 11647  cexp 12816  csqrt 13923   Yrm crmy 36984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-rmy 36986
This theorem is referenced by:  rmxyval  36999
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