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Theorem uzrdgxfr 12177
Description: Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)
Hypotheses
Ref Expression
uzrdgxfr.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  A )  |`  om )
uzrdgxfr.2  |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )
uzrdgxfr.3  |-  A  e.  ZZ
uzrdgxfr.4  |-  B  e.  ZZ
Assertion
Ref Expression
uzrdgxfr  |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    G( x)    H( x)    N( x)

Proof of Theorem uzrdgxfr
Dummy variables  k 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5881 . . 3  |-  ( y  =  (/)  ->  ( G `
 y )  =  ( G `  (/) ) )
2 fveq2 5881 . . . 4  |-  ( y  =  (/)  ->  ( H `
 y )  =  ( H `  (/) ) )
32oveq1d 6320 . . 3  |-  ( y  =  (/)  ->  ( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  (/) )  +  ( A  -  B ) ) )
41, 3eqeq12d 2451 . 2  |-  ( y  =  (/)  ->  ( ( G `  y )  =  ( ( H `
 y )  +  ( A  -  B
) )  <->  ( G `  (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) ) ) )
5 fveq2 5881 . . 3  |-  ( y  =  k  ->  ( G `  y )  =  ( G `  k ) )
6 fveq2 5881 . . . 4  |-  ( y  =  k  ->  ( H `  y )  =  ( H `  k ) )
76oveq1d 6320 . . 3  |-  ( y  =  k  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 k )  +  ( A  -  B
) ) )
85, 7eqeq12d 2451 . 2  |-  ( y  =  k  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  k )  =  ( ( H `  k
)  +  ( A  -  B ) ) ) )
9 fveq2 5881 . . 3  |-  ( y  =  suc  k  -> 
( G `  y
)  =  ( G `
 suc  k )
)
10 fveq2 5881 . . . 4  |-  ( y  =  suc  k  -> 
( H `  y
)  =  ( H `
 suc  k )
)
1110oveq1d 6320 . . 3  |-  ( y  =  suc  k  -> 
( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  suc  k
)  +  ( A  -  B ) ) )
129, 11eqeq12d 2451 . 2  |-  ( y  =  suc  k  -> 
( ( G `  y )  =  ( ( H `  y
)  +  ( A  -  B ) )  <-> 
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) ) ) )
13 fveq2 5881 . . 3  |-  ( y  =  N  ->  ( G `  y )  =  ( G `  N ) )
14 fveq2 5881 . . . 4  |-  ( y  =  N  ->  ( H `  y )  =  ( H `  N ) )
1514oveq1d 6320 . . 3  |-  ( y  =  N  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
1613, 15eqeq12d 2451 . 2  |-  ( y  =  N  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  N )  =  ( ( H `  N
)  +  ( A  -  B ) ) ) )
17 uzrdgxfr.4 . . . . 5  |-  B  e.  ZZ
18 zcn 10942 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1917, 18ax-mp 5 . . . 4  |-  B  e.  CC
20 uzrdgxfr.3 . . . . 5  |-  A  e.  ZZ
21 zcn 10942 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  CC )
2220, 21ax-mp 5 . . . 4  |-  A  e.  CC
2319, 22pncan3i 9950 . . 3  |-  ( B  +  ( A  -  B ) )  =  A
24 uzrdgxfr.2 . . . . 5  |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )
2517, 24om2uz0i 12158 . . . 4  |-  ( H `
 (/) )  =  B
2625oveq1i 6315 . . 3  |-  ( ( H `  (/) )  +  ( A  -  B
) )  =  ( B  +  ( A  -  B ) )
27 uzrdgxfr.1 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  A )  |`  om )
2820, 27om2uz0i 12158 . . 3  |-  ( G `
 (/) )  =  A
2923, 26, 283eqtr4ri 2469 . 2  |-  ( G `
 (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) )
30 oveq1 6312 . . 3  |-  ( ( G `  k )  =  ( ( H `
 k )  +  ( A  -  B
) )  ->  (
( G `  k
)  +  1 )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
3120, 27om2uzsuci 12159 . . . 4  |-  ( k  e.  om  ->  ( G `  suc  k )  =  ( ( G `
 k )  +  1 ) )
3217, 24om2uzsuci 12159 . . . . . 6  |-  ( k  e.  om  ->  ( H `  suc  k )  =  ( ( H `
 k )  +  1 ) )
3332oveq1d 6320 . . . . 5  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  1 )  +  ( A  -  B ) ) )
3417, 24om2uzuzi 12160 . . . . . . . 8  |-  ( k  e.  om  ->  ( H `  k )  e.  ( ZZ>= `  B )
)
35 eluzelz 11168 . . . . . . . 8  |-  ( ( H `  k )  e.  ( ZZ>= `  B
)  ->  ( H `  k )  e.  ZZ )
3634, 35syl 17 . . . . . . 7  |-  ( k  e.  om  ->  ( H `  k )  e.  ZZ )
3736zcnd 11041 . . . . . 6  |-  ( k  e.  om  ->  ( H `  k )  e.  CC )
38 ax-1cn 9596 . . . . . . 7  |-  1  e.  CC
3922, 19subcli 9949 . . . . . . 7  |-  ( A  -  B )  e.  CC
40 add32 9846 . . . . . . 7  |-  ( ( ( H `  k
)  e.  CC  /\  1  e.  CC  /\  ( A  -  B )  e.  CC )  ->  (
( ( H `  k )  +  1 )  +  ( A  -  B ) )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
4138, 39, 40mp3an23 1352 . . . . . 6  |-  ( ( H `  k )  e.  CC  ->  (
( ( H `  k )  +  1 )  +  ( A  -  B ) )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
4237, 41syl 17 . . . . 5  |-  ( k  e.  om  ->  (
( ( H `  k )  +  1 )  +  ( A  -  B ) )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
4333, 42eqtrd 2470 . . . 4  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  ( A  -  B ) )  +  1 ) )
4431, 43eqeq12d 2451 . . 3  |-  ( k  e.  om  ->  (
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) )  <->  ( ( G `
 k )  +  1 )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) ) )
4530, 44syl5ibr 224 . 2  |-  ( k  e.  om  ->  (
( G `  k
)  =  ( ( H `  k )  +  ( A  -  B ) )  -> 
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) ) ) )
464, 8, 12, 16, 29, 45finds 6733 1  |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767    |-> cmpt 4484    |` cres 4856   suc csuc 5444   ` cfv 5601  (class class class)co 6305   omcom 6706   reccrdg 7135   CCcc 9536   1c1 9539    + caddc 9541    - cmin 9859   ZZcz 10937   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160
This theorem is referenced by:  fz1isolem  12619
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