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Theorem rmxfval 37994
Description: Value of the X sequence. Not used after rmxyval 38006 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 6803 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎↑2) = (𝐴↑2))
21fvoveq1d 6818 . . . . . . . . 9 (𝑎 = 𝐴 → (√‘((𝑎↑2) − 1)) = (√‘((𝐴↑2) − 1)))
32oveq1d 6811 . . . . . . . 8 (𝑎 = 𝐴 → ((√‘((𝑎↑2) − 1)) · (2nd𝑏)) = ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))
43oveq2d 6812 . . . . . . 7 (𝑎 = 𝐴 → ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))) = ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))
54mpteq2dv 4880 . . . . . 6 (𝑎 = 𝐴 → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
65cnveqd 5435 . . . . 5 (𝑎 = 𝐴(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏)))) = (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))))
76adantr 466 . . . 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 6814 . . . . 5 (𝑎 = 𝐴 → (𝑎 + (√‘((𝑎↑2) − 1))) = (𝐴 + (√‘((𝐴↑2) − 1))))
10 id 22 . . . . 5 (𝑛 = 𝑁𝑛 = 𝑁)
119, 10oveqan12d 6815 . . . 4 ((𝑎 = 𝐴𝑛 = 𝑁) → ((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
127, 11fveq12d 6340 . . 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 6337 . 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))))
14 df-rmx 37992 . 2 Xrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
15 fvex 6344 . 2 (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))) ∈ V
1613, 14, 15ovmpt2a 6942 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 382   = wceq 1631  wcel 2145  cmpt 4864   × cxp 5248  ccnv 5249  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  1c1 10143   + caddc 10145   · cmul 10147  cmin 10472  2c2 11276  0cn0 11499  cz 11584  cuz 11893  cexp 13067  csqrt 14181   Xrm crmx 37990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-rmx 37992
This theorem is referenced by:  rmxyval  38006
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