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Theorem rmyfval 31080
Description: Value of the Y sequence. Not used after rmxyval 31090 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmyfval  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A Yrm 
N )  =  ( 2nd `  ( `' ( 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 rmyfval
Dummy variables  n  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6277 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
21oveq1d 6285 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a ^ 2 )  -  1 )  =  ( ( A ^ 2 )  - 
1 ) )
32fveq2d 5852 . . . . . . . . 9  |-  ( a  =  A  ->  ( sqr `  ( ( a ^ 2 )  - 
1 ) )  =  ( sqr `  (
( A ^ 2 )  -  1 ) ) )
43oveq1d 6285 . . . . . . . 8  |-  ( a  =  A  ->  (
( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )
54oveq2d 6286 . . . . . . 7  |-  ( a  =  A  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
65mpteq2dv 4526 . . . . . 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 5167 . . . . 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 463 . . . 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 22 . . . . . 6  |-  ( a  =  A  ->  a  =  A )
109, 3oveq12d 6288 . . . . 5  |-  ( a  =  A  ->  (
a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) )
11 id 22 . . . . 5  |-  ( n  =  N  ->  n  =  N )
1210, 11oveqan12d 6289 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  ( ( a  +  ( sqr `  (
( a ^ 2 )  -  1 ) ) ) ^ n
)  =  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) )
138, 12fveq12d 5854 . . 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 5852 . 2  |-  ( ( a  =  A  /\  n  =  N )  ->  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) ) ^
n ) ) )  =  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) ) ) )
15 df-rmy 31078 . 2  |- Yrm  =  (
a  e.  ( ZZ>= ` 
2 ) ,  n  e.  ZZ  |->  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  ( ( a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) ) ^
n ) ) ) )
16 fvex 5858 . 2  |-  ( 2nd `  ( `' ( 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 6406 1  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A Yrm 
N )  =  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ^cexp 12148   sqrcsqrt 13148   Yrm crmy 31076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-rmy 31078
This theorem is referenced by:  rmxyval  31090
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