Theorem uzrdgxfr 12766

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( y = (/) -> ( G ` y ) = ( G ` (/) ) )
2		fveq2 6191	. . . 4 |- ( y = (/) -> ( H ` y ) = ( H ` (/) ) )
3	2	oveq1d 6665	. . 3 |- ( y = (/) -> ( ( H ` y ) + ( A - B ) ) = ( ( H ` (/) ) + ( A - B ) ) )
4	1, 3	eqeq12d 2637	. 2 |- ( y = (/) -> ( ( G ` y ) = ( ( H ` y ) + ( A - B ) ) <-> ( G ` (/) ) = ( ( H ` (/) ) + ( A - B ) ) ) )
5		fveq2 6191	. . 3 |- ( y = k -> ( G ` y ) = ( G ` k ) )
6		fveq2 6191	. . . 4 |- ( y = k -> ( H ` y ) = ( H ` k ) )
7	6	oveq1d 6665	. . 3 |- ( y = k -> ( ( H ` y ) + ( A - B ) ) = ( ( H ` k ) + ( A - B ) ) )
8	5, 7	eqeq12d 2637	. 2 |- ( y = k -> ( ( G ` y ) = ( ( H ` y ) + ( A - B ) ) <-> ( G ` k ) = ( ( H ` k ) + ( A - B ) ) ) )
9		fveq2 6191	. . 3 |- ( y = suc k -> ( G ` y ) = ( G ` suc k ) )
10		fveq2 6191	. . . 4 |- ( y = suc k -> ( H ` y ) = ( H ` suc k ) )
11	10	oveq1d 6665	. . 3 |- ( y = suc k -> ( ( H ` y ) + ( A - B ) ) = ( ( H ` suc k ) + ( A - B ) ) )
12	9, 11	eqeq12d 2637	. 2 |- ( y = suc k -> ( ( G ` y ) = ( ( H ` y ) + ( A - B ) ) <-> ( G ` suc k ) = ( ( H ` suc k ) + ( A - B ) ) ) )
13		fveq2 6191	. . 3 |- ( y = N -> ( G ` y ) = ( G ` N ) )
14		fveq2 6191	. . . 4 |- ( y = N -> ( H ` y ) = ( H ` N ) )
15	14	oveq1d 6665	. . 3 |- ( y = N -> ( ( H ` y ) + ( A - B ) ) = ( ( H ` N ) + ( A - B ) ) )
16	13, 15	eqeq12d 2637	. 2 |- ( y = N -> ( ( G ` y ) = ( ( H ` y ) + ( A - B ) ) <-> ( G ` N ) = ( ( H ` N ) + ( A - B ) ) ) )
17		uzrdgxfr.4	. . . . 5 |- B e. ZZ
18		zcn 11382	. . . . 5 |- ( B e. ZZ -> B e. CC )
19	17, 18	ax-mp 5	. . . 4 |- B e. CC
20		uzrdgxfr.3	. . . . 5 |- A e. ZZ
21		zcn 11382	. . . . 5 |- ( A e. ZZ -> A e. CC )
22	20, 21	ax-mp 5	. . . 4 |- A e. CC
23	19, 22	pncan3i 10358	. . 3 |- ( B + ( A - B ) ) = A
24		uzrdgxfr.2	. . . . 5 |- H = ( rec ( ( x e. _V |-> ( x + 1 ) ) , B ) |` om )
25	17, 24	om2uz0i 12746	. . . 4 |- ( H ` (/) ) = B
26	25	oveq1i 6660	. . 3 |- ( ( H ` (/) ) + ( A - B ) ) = ( B + ( A - B ) )
27		uzrdgxfr.1	. . . 4 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , A ) |` om )
28	20, 27	om2uz0i 12746	. . 3 |- ( G ` (/) ) = A
29	23, 26, 28	3eqtr4ri 2655	. 2 |- ( G ` (/) ) = ( ( H ` (/) ) + ( A - B ) )
30		oveq1 6657	. . 3 |- ( ( G ` k ) = ( ( H ` k ) + ( A - B ) ) -> ( ( G ` k ) + 1 ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) )
31	20, 27	om2uzsuci 12747	. . . 4 |- ( k e. om -> ( G ` suc k ) = ( ( G ` k ) + 1 ) )
32	17, 24	om2uzsuci 12747	. . . . . 6 |- ( k e. om -> ( H ` suc k ) = ( ( H ` k ) + 1 ) )
33	32	oveq1d 6665	. . . . 5 |- ( k e. om -> ( ( H ` suc k ) + ( A - B ) ) = ( ( ( H ` k ) + 1 ) + ( A - B ) ) )
34	17, 24	om2uzuzi 12748	. . . . . . . 8 |- ( k e. om -> ( H ` k ) e. ( ZZ>= ` B ) )
35		eluzelz 11697	. . . . . . . 8 |- ( ( H ` k ) e. ( ZZ>= ` B ) -> ( H ` k ) e. ZZ )
36	34, 35	syl 17	. . . . . . 7 |- ( k e. om -> ( H ` k ) e. ZZ )
37	36	zcnd 11483	. . . . . 6 |- ( k e. om -> ( H ` k ) e. CC )
38		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
39	22, 19	subcli 10357	. . . . . . 7 |- ( A - B ) e. CC
40		add32 10254	. . . . . . 7 |- ( ( ( H ` k ) e. CC /\ 1 e. CC /\ ( A - B ) e. CC ) -> ( ( ( H ` k ) + 1 ) + ( A - B ) ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) )
41	38, 39, 40	mp3an23 1416	. . . . . 6 |- ( ( H ` k ) e. CC -> ( ( ( H ` k ) + 1 ) + ( A - B ) ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) )
42	37, 41	syl 17	. . . . 5 |- ( k e. om -> ( ( ( H ` k ) + 1 ) + ( A - B ) ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) )
43	33, 42	eqtrd 2656	. . . 4 |- ( k e. om -> ( ( H ` suc k ) + ( A - B ) ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) )
44	31, 43	eqeq12d 2637	. . 3 |- ( k e. om -> ( ( G ` suc k ) = ( ( H ` suc k ) + ( A - B ) ) <-> ( ( G ` k ) + 1 ) = ( ( ( H ` k ) + ( A - B ) ) + 1 ) ) )
45	30, 44	syl5ibr 236	. 2 |- ( k e. om -> ( ( G ` k ) = ( ( H ` k ) + ( A - B ) ) -> ( G ` suc k ) = ( ( H ` suc k ) + ( A - B ) ) ) )
46	4, 8, 12, 16, 29, 45	finds 7092	1 |- ( N e. om -> ( G ` N ) = ( ( H ` N ) + ( A - B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   CCcc 9934   1c1 9937   + caddc 9939   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  fz1isolem  13245