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Theorem uzrdgxfr 11899
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 5792 . . 3  |-  ( y  =  (/)  ->  ( G `
 y )  =  ( G `  (/) ) )
2 fveq2 5792 . . . 4  |-  ( y  =  (/)  ->  ( H `
 y )  =  ( H `  (/) ) )
32oveq1d 6208 . . 3  |-  ( y  =  (/)  ->  ( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  (/) )  +  ( A  -  B ) ) )
41, 3eqeq12d 2473 . 2  |-  ( y  =  (/)  ->  ( ( G `  y )  =  ( ( H `
 y )  +  ( A  -  B
) )  <->  ( G `  (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) ) ) )
5 fveq2 5792 . . 3  |-  ( y  =  k  ->  ( G `  y )  =  ( G `  k ) )
6 fveq2 5792 . . . 4  |-  ( y  =  k  ->  ( H `  y )  =  ( H `  k ) )
76oveq1d 6208 . . 3  |-  ( y  =  k  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 k )  +  ( A  -  B
) ) )
85, 7eqeq12d 2473 . 2  |-  ( y  =  k  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  k )  =  ( ( H `  k
)  +  ( A  -  B ) ) ) )
9 fveq2 5792 . . 3  |-  ( y  =  suc  k  -> 
( G `  y
)  =  ( G `
 suc  k )
)
10 fveq2 5792 . . . 4  |-  ( y  =  suc  k  -> 
( H `  y
)  =  ( H `
 suc  k )
)
1110oveq1d 6208 . . 3  |-  ( y  =  suc  k  -> 
( ( H `  y )  +  ( A  -  B ) )  =  ( ( H `  suc  k
)  +  ( A  -  B ) ) )
129, 11eqeq12d 2473 . 2  |-  ( y  =  suc  k  -> 
( ( G `  y )  =  ( ( H `  y
)  +  ( A  -  B ) )  <-> 
( G `  suc  k )  =  ( ( H `  suc  k )  +  ( A  -  B ) ) ) )
13 fveq2 5792 . . 3  |-  ( y  =  N  ->  ( G `  y )  =  ( G `  N ) )
14 fveq2 5792 . . . 4  |-  ( y  =  N  ->  ( H `  y )  =  ( H `  N ) )
1514oveq1d 6208 . . 3  |-  ( y  =  N  ->  (
( H `  y
)  +  ( A  -  B ) )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
1613, 15eqeq12d 2473 . 2  |-  ( y  =  N  ->  (
( G `  y
)  =  ( ( H `  y )  +  ( A  -  B ) )  <->  ( G `  N )  =  ( ( H `  N
)  +  ( A  -  B ) ) ) )
17 uzrdgxfr.4 . . . . 5  |-  B  e.  ZZ
18 zcn 10755 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1917, 18ax-mp 5 . . . 4  |-  B  e.  CC
20 uzrdgxfr.3 . . . . 5  |-  A  e.  ZZ
21 zcn 10755 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  CC )
2220, 21ax-mp 5 . . . 4  |-  A  e.  CC
2319, 22pncan3i 9789 . . 3  |-  ( B  +  ( A  -  B ) )  =  A
24 uzrdgxfr.2 . . . . 5  |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )
2517, 24om2uz0i 11880 . . . 4  |-  ( H `
 (/) )  =  B
2625oveq1i 6203 . . 3  |-  ( ( H `  (/) )  +  ( A  -  B
) )  =  ( B  +  ( A  -  B ) )
27 uzrdgxfr.1 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  A )  |`  om )
2820, 27om2uz0i 11880 . . 3  |-  ( G `
 (/) )  =  A
2923, 26, 283eqtr4ri 2491 . 2  |-  ( G `
 (/) )  =  ( ( H `  (/) )  +  ( A  -  B
) )
30 oveq1 6200 . . 3  |-  ( ( G `  k )  =  ( ( H `
 k )  +  ( A  -  B
) )  ->  (
( G `  k
)  +  1 )  =  ( ( ( H `  k )  +  ( A  -  B ) )  +  1 ) )
3120, 27om2uzsuci 11881 . . . 4  |-  ( k  e.  om  ->  ( G `  suc  k )  =  ( ( G `
 k )  +  1 ) )
3217, 24om2uzsuci 11881 . . . . . 6  |-  ( k  e.  om  ->  ( H `  suc  k )  =  ( ( H `
 k )  +  1 ) )
3332oveq1d 6208 . . . . 5  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  1 )  +  ( A  -  B ) ) )
3417, 24om2uzuzi 11882 . . . . . . . 8  |-  ( k  e.  om  ->  ( H `  k )  e.  ( ZZ>= `  B )
)
35 eluzelz 10974 . . . . . . . 8  |-  ( ( H `  k )  e.  ( ZZ>= `  B
)  ->  ( H `  k )  e.  ZZ )
3634, 35syl 16 . . . . . . 7  |-  ( k  e.  om  ->  ( H `  k )  e.  ZZ )
3736zcnd 10852 . . . . . 6  |-  ( k  e.  om  ->  ( H `  k )  e.  CC )
38 ax-1cn 9444 . . . . . . 7  |-  1  e.  CC
3922, 19subcli 9788 . . . . . . 7  |-  ( A  -  B )  e.  CC
40 add32 9687 . . . . . . 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 1307 . . . . . 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 2492 . . . 4  |-  ( k  e.  om  ->  (
( H `  suc  k )  +  ( A  -  B ) )  =  ( ( ( H `  k
)  +  ( A  -  B ) )  +  1 ) )
4431, 43eqeq12d 2473 . . 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 6605 1  |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `
 N )  +  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3071   (/)c0 3738    |-> cmpt 4451   suc csuc 4822    |` cres 4943   ` cfv 5519  (class class class)co 6193   omcom 6579   reccrdg 6968   CCcc 9384   1c1 9387    + caddc 9389    - cmin 9699   ZZcz 10750   ZZ>=cuz 10965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966
This theorem is referenced by:  fz1isolem  12325
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