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Theorem rmxfval 36934
Description: Value of the X sequence. Not used after rmxyval 36946 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) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((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.
StepHypRef Expression
1 oveq1 6612 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
21oveq1d 6620 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑎↑2) − 1) = ((𝐴↑2) − 1))
32fveq2d 6154 . . . . . . . . 9 (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
43oveq1d 6620 . . . . . . . 8 (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))
54oveq2d 6621 . . . . . . 7 (𝑎 = 𝐴 → ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))) = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
65mpteq2dv 4710 . . . . . 6 (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
76cnveqd 5263 . . . . 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 6623 . . . . 5 (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
11 id 22 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
1210, 11oveqan12d 6624 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
138, 12fveq12d 6156 . . 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 6154 . 2 ((𝑎 = 𝐴𝑛 = 𝑁) → (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))) = (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
15 df-rmx 36932 . 2 Xrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
16 fvex 6160 . 2 (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
1714, 15, 16ovmpt2a 6745 1 ((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘((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 1992  cmpt 4678   × cxp 5077  ccnv 5078  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  1c1 9882   + caddc 9884   · cmul 9886  cmin 10211  2c2 11015  0cn0 11237  cz 11322  cuz 11631  cexp 12797  csqrt 13902   Xrm crmx 36930
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-rmx 36932
This theorem is referenced by:  rmxyval  36946
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