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Theorem rmyfval 29272
Description: Value of the Y sequence. Not used after rmxyval 29282 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 6119 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
21oveq1d 6127 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a ^ 2 )  -  1 )  =  ( ( A ^ 2 )  - 
1 ) )
32fveq2d 5716 . . . . . . . . 9  |-  ( a  =  A  ->  ( sqr `  ( ( a ^ 2 )  - 
1 ) )  =  ( sqr `  (
( A ^ 2 )  -  1 ) ) )
43oveq1d 6127 . . . . . . . 8  |-  ( a  =  A  ->  (
( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )
54oveq2d 6128 . . . . . . 7  |-  ( a  =  A  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
65mpteq2dv 4400 . . . . . 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 5036 . . . . 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 465 . . . 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 6130 . . . . 5  |-  ( a  =  A  ->  (
a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) )
11 id 22 . . . . 5  |-  ( n  =  N  ->  n  =  N )
1210, 11oveqan12d 6131 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  ( ( a  +  ( sqr `  (
( a ^ 2 )  -  1 ) ) ) ^ n
)  =  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) )
138, 12fveq12d 5718 . . 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 5716 . 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 29270 . 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 5722 . 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 6242 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 369    = wceq 1369    e. wcel 1756    e. cmpt 4371    X. cxp 4859   `'ccnv 4860   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597   1c1 9304    + caddc 9306    x. cmul 9308    - cmin 9616   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882   ^cexp 11886   sqrcsqr 12743   Yrm crmy 29268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-rmy 29270
This theorem is referenced by:  rmxyval  29282
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