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

Theorem rmxfval 26968
Description: Value of the X sequence. Not used after rmxyval 26979 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmxfval  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A Xrm 
N )  =  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 ( ( A  +  ( sqr `  (
( 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 6089 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
21oveq1d 6097 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a ^ 2 )  -  1 )  =  ( ( A ^ 2 )  - 
1 ) )
32fveq2d 5733 . . . . . . . . 9  |-  ( a  =  A  ->  ( sqr `  ( ( a ^ 2 )  - 
1 ) )  =  ( sqr `  (
( A ^ 2 )  -  1 ) ) )
43oveq1d 6097 . . . . . . . 8  |-  ( a  =  A  ->  (
( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )
54oveq2d 6098 . . . . . . 7  |-  ( a  =  A  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
65mpteq2dv 4297 . . . . . 6  |-  ( a  =  A  ->  (
b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )  =  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) )
76cnveqd 5049 . . . . 5  |-  ( a  =  A  ->  `' ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )  =  `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) )
87adantr 453 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  =  `' ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) )
9 id 21 . . . . . 6  |-  ( a  =  A  ->  a  =  A )
109, 3oveq12d 6100 . . . . 5  |-  ( a  =  A  ->  (
a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) )
11 id 21 . . . . 5  |-  ( n  =  N  ->  n  =  N )
1210, 11oveqan12d 6101 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  ( ( a  +  ( sqr `  (
( a ^ 2 )  -  1 ) ) ) ^ n
)  =  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) )
138, 12fveq12d 5735 . . 3  |-  ( ( a  =  A  /\  n  =  N )  ->  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( a ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  (
( a  +  ( sqr `  ( ( a ^ 2 )  -  1 ) ) ) ^ n ) )  =  ( `' ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
) ) )
1413fveq2d 5733 . 2  |-  ( ( a  =  A  /\  n  =  N )  ->  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) ) ^
n ) ) )  =  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) ) ) )
15 df-rmx 26966 . 2  |- Xrm  =  (
a  e.  ( ZZ>= ` 
2 ) ,  n  e.  ZZ  |->  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) ) ^
n ) ) ) )
16 fvex 5743 . 2  |-  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N ) ) )  e.  _V
1714, 15, 16ovmpt2a 6205 1  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A Xrm 
N )  =  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4267    X. cxp 4877   `'ccnv 4878   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   1c1 8992    + caddc 8994    x. cmul 8996    - cmin 9292   2c2 10050   NN0cn0 10222   ZZcz 10283   ZZ>=cuz 10489   ^cexp 11383   sqrcsqr 12039   Xrm crmx 26964
This theorem is referenced by:  rmxyval  26979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-rmx 26966
  Copyright terms: Public domain W3C validator