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Theorem uzrdgxfr 12045
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 5866 . . 3  |-  ( y  =  (/)  ->  ( G `
 y )  =  ( G `  (/) ) )
2 fveq2 5866 . . . 4  |-  ( y  =  (/)  ->  ( H `
 y )  =  ( H `  (/) ) )
32oveq1d 6299 . . 3  |-  ( y  =  (/)  ->  ( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  (/) )  +  ( A  -  B ) ) )
41, 3eqeq12d 2489 . 2  |-  ( y  =  (/)  ->  ( ( G `  y )  =  ( ( H `
 y )  +  ( A  -  B
) )  <->  ( G `  (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) ) ) )
5 fveq2 5866 . . 3  |-  ( y  =  k  ->  ( G `  y )  =  ( G `  k ) )
6 fveq2 5866 . . . 4  |-  ( y  =  k  ->  ( H `  y )  =  ( H `  k ) )
76oveq1d 6299 . . 3  |-  ( y  =  k  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 k )  +  ( A  -  B
) ) )
85, 7eqeq12d 2489 . 2  |-  ( y  =  k  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  k )  =  ( ( H `  k
)  +  ( A  -  B ) ) ) )
9 fveq2 5866 . . 3  |-  ( y  =  suc  k  -> 
( G `  y
)  =  ( G `
 suc  k )
)
10 fveq2 5866 . . . 4  |-  ( y  =  suc  k  -> 
( H `  y
)  =  ( H `
 suc  k )
)
1110oveq1d 6299 . . 3  |-  ( y  =  suc  k  -> 
( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  suc  k
)  +  ( A  -  B ) ) )
129, 11eqeq12d 2489 . 2  |-  ( y  =  suc  k  -> 
( ( G `  y )  =  ( ( H `  y
)  +  ( A  -  B ) )  <-> 
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) ) ) )
13 fveq2 5866 . . 3  |-  ( y  =  N  ->  ( G `  y )  =  ( G `  N ) )
14 fveq2 5866 . . . 4  |-  ( y  =  N  ->  ( H `  y )  =  ( H `  N ) )
1514oveq1d 6299 . . 3  |-  ( y  =  N  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
1613, 15eqeq12d 2489 . 2  |-  ( y  =  N  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  N )  =  ( ( H `  N
)  +  ( A  -  B ) ) ) )
17 uzrdgxfr.4 . . . . 5  |-  B  e.  ZZ
18 zcn 10869 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1917, 18ax-mp 5 . . . 4  |-  B  e.  CC
20 uzrdgxfr.3 . . . . 5  |-  A  e.  ZZ
21 zcn 10869 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  CC )
2220, 21ax-mp 5 . . . 4  |-  A  e.  CC
2319, 22pncan3i 9896 . . 3  |-  ( B  +  ( A  -  B ) )  =  A
24 uzrdgxfr.2 . . . . 5  |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )
2517, 24om2uz0i 12026 . . . 4  |-  ( H `
 (/) )  =  B
2625oveq1i 6294 . . 3  |-  ( ( H `  (/) )  +  ( A  -  B
) )  =  ( B  +  ( A  -  B ) )
27 uzrdgxfr.1 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  A )  |`  om )
2820, 27om2uz0i 12026 . . 3  |-  ( G `
 (/) )  =  A
2923, 26, 283eqtr4ri 2507 . 2  |-  ( G `
 (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) )
30 oveq1 6291 . . 3  |-  ( ( G `  k )  =  ( ( H `
 k )  +  ( A  -  B
) )  ->  (
( G `  k
)  +  1 )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
3120, 27om2uzsuci 12027 . . . 4  |-  ( k  e.  om  ->  ( G `  suc  k )  =  ( ( G `
 k )  +  1 ) )
3217, 24om2uzsuci 12027 . . . . . 6  |-  ( k  e.  om  ->  ( H `  suc  k )  =  ( ( H `
 k )  +  1 ) )
3332oveq1d 6299 . . . . 5  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  1 )  +  ( A  -  B ) ) )
3417, 24om2uzuzi 12028 . . . . . . . 8  |-  ( k  e.  om  ->  ( H `  k )  e.  ( ZZ>= `  B )
)
35 eluzelz 11091 . . . . . . . 8  |-  ( ( H `  k )  e.  ( ZZ>= `  B
)  ->  ( H `  k )  e.  ZZ )
3634, 35syl 16 . . . . . . 7  |-  ( k  e.  om  ->  ( H `  k )  e.  ZZ )
3736zcnd 10967 . . . . . 6  |-  ( k  e.  om  ->  ( H `  k )  e.  CC )
38 ax-1cn 9550 . . . . . . 7  |-  1  e.  CC
3922, 19subcli 9895 . . . . . . 7  |-  ( A  -  B )  e.  CC
40 add32 9793 . . . . . . 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 1316 . . . . . 6  |-  ( ( H `  k )  e.  CC  ->  (
( ( H `  k )  +  1 )  +  ( A  -  B ) )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
4237, 41syl 16 . . . . 5  |-  ( k  e.  om  ->  (
( ( H `  k )  +  1 )  +  ( A  -  B ) )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
4333, 42eqtrd 2508 . . . 4  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  ( A  -  B ) )  +  1 ) )
4431, 43eqeq12d 2489 . . 3  |-  ( k  e.  om  ->  (
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) )  <->  ( ( G `
 k )  +  1 )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) ) )
4530, 44syl5ibr 221 . 2  |-  ( k  e.  om  ->  (
( G `  k
)  =  ( ( H `  k )  +  ( A  -  B ) )  -> 
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) ) ) )
464, 8, 12, 16, 29, 45finds 6710 1  |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785    |-> cmpt 4505   suc csuc 4880    |` cres 5001   ` cfv 5588  (class class class)co 6284   omcom 6684   reccrdg 7075   CCcc 9490   1c1 9493    + caddc 9495    - cmin 9805   ZZcz 10864   ZZ>=cuz 11082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083
This theorem is referenced by:  fz1isolem  12476
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