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Theorem rmyfval 26970
Description: Value of the Y sequence. Not used after rmxyval 26980 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 6090 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
21oveq1d 6098 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a ^ 2 )  -  1 )  =  ( ( A ^ 2 )  - 
1 ) )
32fveq2d 5734 . . . . . . . . 9  |-  ( a  =  A  ->  ( sqr `  ( ( a ^ 2 )  - 
1 ) )  =  ( sqr `  (
( A ^ 2 )  -  1 ) ) )
43oveq1d 6098 . . . . . . . 8  |-  ( a  =  A  ->  (
( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )
54oveq2d 6099 . . . . . . 7  |-  ( a  =  A  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
65mpteq2dv 4298 . . . . . 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 5050 . . . . 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 6101 . . . . 5  |-  ( a  =  A  ->  (
a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) )
11 id 21 . . . . 5  |-  ( n  =  N  ->  n  =  N )
1210, 11oveqan12d 6102 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  ( ( a  +  ( sqr `  (
( a ^ 2 )  -  1 ) ) ) ^ n
)  =  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) )
138, 12fveq12d 5736 . . 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 5734 . 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 26968 . 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 5744 . 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 6206 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268    X. cxp 4878   `'ccnv 4879   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ^cexp 11384   sqrcsqr 12040   Yrm crmy 26966
This theorem is referenced by:  rmxyval  26980
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 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-rmy 26968
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