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Theorem rmxfval 29242
Description: Value of the X sequence. Not used after rmxyval 29253 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 6096 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a ^ 2 )  =  ( A ^
2 ) )
21oveq1d 6104 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a ^ 2 )  -  1 )  =  ( ( A ^ 2 )  - 
1 ) )
32fveq2d 5693 . . . . . . . . 9  |-  ( a  =  A  ->  ( sqr `  ( ( a ^ 2 )  - 
1 ) )  =  ( sqr `  (
( A ^ 2 )  -  1 ) ) )
43oveq1d 6104 . . . . . . . 8  |-  ( a  =  A  ->  (
( sqr `  (
( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )
54oveq2d 6105 . . . . . . 7  |-  ( a  =  A  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
65mpteq2dv 4377 . . . . . 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 5013 . . . . 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 6107 . . . . 5  |-  ( a  =  A  ->  (
a  +  ( sqr `  ( ( a ^
2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) )
11 id 22 . . . . 5  |-  ( n  =  N  ->  n  =  N )
1210, 11oveqan12d 6108 . . . 4  |-  ( ( a  =  A  /\  n  =  N )  ->  ( ( a  +  ( sqr `  (
( a ^ 2 )  -  1 ) ) ) ^ n
)  =  ( ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) ^ N ) )
138, 12fveq12d 5695 . . 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 5693 . 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 29240 . 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 5699 . 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 6219 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574   1c1 9281    + caddc 9283    x. cmul 9285    - cmin 9593   2c2 10369   NN0cn0 10577   ZZcz 10644   ZZ>=cuz 10859   ^cexp 11863   sqrcsqr 12720   Xrm crmx 29238
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 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-rmx 29240
This theorem is referenced by:  rmxyval  29253
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