Theorem uzrdgsuci 12759

Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Hypotheses

Ref	Expression
om2uz.1	|- C e. ZZ
om2uz.2	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
uzrdg.1	|- A e. _V
uzrdg.2	|- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )
uzrdg.3	|- S = ran R

Assertion

Ref	Expression
uzrdgsuci	|- ( B e. ( ZZ>= ` C ) -> ( S ` ( B + 1 ) ) = ( B F ( S ` B ) ) )

Distinct variable groups:   y, A   x, y, C   y, G   x, F, y

Allowed substitution hints:   A( x)   B( x, y)   R( x, y)   S( x, y)   G( x)

Proof of Theorem uzrdgsuci

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		om2uz.1	. . . . . 6 |- C e. ZZ
2		om2uz.2	. . . . . 6 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
3		uzrdg.1	. . . . . 6 |- A e. _V
4		uzrdg.2	. . . . . 6 |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )
5		uzrdg.3	. . . . . 6 |- S = ran R
6	1, 2, 3, 4, 5	uzrdgfni 12757	. . . . 5 |- S Fn ( ZZ>= ` C )
7		fnfun 5988	. . . . 5 |- ( S Fn ( ZZ>= ` C ) -> Fun S )
8	6, 7	ax-mp 5	. . . 4 |- Fun S
9		peano2uz 11741	. . . . . 6 |- ( B e. ( ZZ>= ` C ) -> ( B + 1 ) e. ( ZZ>= ` C ) )
10	1, 2, 3, 4	uzrdglem 12756	. . . . . 6 |- ( ( B + 1 ) e. ( ZZ>= ` C ) -> <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. ran R )
11	9, 10	syl 17	. . . . 5 |- ( B e. ( ZZ>= ` C ) -> <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. ran R )
12	11, 5	syl6eleqr 2712	. . . 4 |- ( B e. ( ZZ>= ` C ) -> <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. S )
13		funopfv 6235	. . . 4 |- ( Fun S -> ( <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. S -> ( S ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) ) )
14	8, 12, 13	mpsyl 68	. . 3 |- ( B e. ( ZZ>= ` C ) -> ( S ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) )
15	1, 2	om2uzf1oi 12752	. . . . . . . 8 |- G : om -1-1-onto-> ( ZZ>= ` C )
16		f1ocnvdm 6540	. . . . . . . 8 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
17	15, 16	mpan 706	. . . . . . 7 |- ( B e. ( ZZ>= ` C ) -> ( `' G ` B ) e. om )
18		peano2 7086	. . . . . . 7 |- ( ( `' G ` B ) e. om -> suc ( `' G ` B ) e. om )
19	17, 18	syl 17	. . . . . 6 |- ( B e. ( ZZ>= ` C ) -> suc ( `' G ` B ) e. om )
20	1, 2	om2uzsuci 12747	. . . . . . . 8 |- ( ( `' G ` B ) e. om -> ( G ` suc ( `' G ` B ) ) = ( ( G ` ( `' G ` B ) ) + 1 ) )
21	17, 20	syl 17	. . . . . . 7 |- ( B e. ( ZZ>= ` C ) -> ( G ` suc ( `' G ` B ) ) = ( ( G ` ( `' G ` B ) ) + 1 ) )
22		f1ocnvfv2 6533	. . . . . . . . 9 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) = B )
23	15, 22	mpan 706	. . . . . . . 8 |- ( B e. ( ZZ>= ` C ) -> ( G ` ( `' G ` B ) ) = B )
24	23	oveq1d 6665	. . . . . . 7 |- ( B e. ( ZZ>= ` C ) -> ( ( G ` ( `' G ` B ) ) + 1 ) = ( B + 1 ) )
25	21, 24	eqtrd 2656	. . . . . 6 |- ( B e. ( ZZ>= ` C ) -> ( G ` suc ( `' G ` B ) ) = ( B + 1 ) )
26		f1ocnvfv 6534	. . . . . . 7 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ suc ( `' G ` B ) e. om ) -> ( ( G ` suc ( `' G ` B ) ) = ( B + 1 ) -> ( `' G ` ( B + 1 ) ) = suc ( `' G ` B ) ) )
27	15, 26	mpan 706	. . . . . 6 |- ( suc ( `' G ` B ) e. om -> ( ( G ` suc ( `' G ` B ) ) = ( B + 1 ) -> ( `' G ` ( B + 1 ) ) = suc ( `' G ` B ) ) )
28	19, 25, 27	sylc 65	. . . . 5 |- ( B e. ( ZZ>= ` C ) -> ( `' G ` ( B + 1 ) ) = suc ( `' G ` B ) )
29	28	fveq2d 6195	. . . 4 |- ( B e. ( ZZ>= ` C ) -> ( R ` ( `' G ` ( B + 1 ) ) ) = ( R ` suc ( `' G ` B ) ) )
30	29	fveq2d 6195	. . 3 |- ( B e. ( ZZ>= ` C ) -> ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) = ( 2nd ` ( R ` suc ( `' G ` B ) ) ) )
31	14, 30	eqtrd 2656	. 2 |- ( B e. ( ZZ>= ` C ) -> ( S ` ( B + 1 ) ) = ( 2nd ` ( R ` suc ( `' G ` B ) ) ) )
32		frsuc 7532	. . . . . . . 8 |- ( ( `' G ` B ) e. om -> ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` suc ( `' G ` B ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` ( `' G ` B ) ) ) )
33	4	fveq1i 6192	. . . . . . . 8 |- ( R ` suc ( `' G ` B ) ) = ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` suc ( `' G ` B ) )
34	4	fveq1i 6192	. . . . . . . . 9 |- ( R ` ( `' G ` B ) ) = ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` ( `' G ` B ) )
35	34	fveq2i 6194	. . . . . . . 8 |- ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` ( `' G ` B ) ) )
36	32, 33, 35	3eqtr4g 2681	. . . . . . 7 |- ( ( `' G ` B ) e. om -> ( R ` suc ( `' G ` B ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) )
37	1, 2, 3, 4	om2uzrdg 12755	. . . . . . . . 9 |- ( ( `' G ` B ) e. om -> ( R ` ( `' G ` B ) ) = <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
38	37	fveq2d 6195	. . . . . . . 8 |- ( ( `' G ` B ) e. om -> ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. ) )
39		df-ov 6653	. . . . . . . 8 |- ( ( G ` ( `' G ` B ) ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
40	38, 39	syl6eqr 2674	. . . . . . 7 |- ( ( `' G ` B ) e. om -> ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( G ` ( `' G ` B ) ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
41	36, 40	eqtrd 2656	. . . . . 6 |- ( ( `' G ` B ) e. om -> ( R ` suc ( `' G ` B ) ) = ( ( G ` ( `' G ` B ) ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
42		fvex 6201	. . . . . . 7 |- ( G ` ( `' G ` B ) ) e. _V
43		fvex 6201	. . . . . . 7 |- ( 2nd ` ( R ` ( `' G ` B ) ) ) e. _V
44		oveq1 6657	. . . . . . . . 9 |- ( z = ( G ` ( `' G ` B ) ) -> ( z + 1 ) = ( ( G ` ( `' G ` B ) ) + 1 ) )
45		oveq1 6657	. . . . . . . . 9 |- ( z = ( G ` ( `' G ` B ) ) -> ( z F w ) = ( ( G ` ( `' G ` B ) ) F w ) )
46	44, 45	opeq12d 4410	. . . . . . . 8 |- ( z = ( G ` ( `' G ` B ) ) -> <. ( z + 1 ) , ( z F w ) >. = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F w ) >. )
47		oveq2 6658	. . . . . . . . 9 |- ( w = ( 2nd ` ( R ` ( `' G ` B ) ) ) -> ( ( G ` ( `' G ` B ) ) F w ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
48	47	opeq2d 4409	. . . . . . . 8 |- ( w = ( 2nd ` ( R ` ( `' G ` B ) ) ) -> <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F w ) >. = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
49		oveq1 6657	. . . . . . . . . 10 |- ( x = z -> ( x + 1 ) = ( z + 1 ) )
50		oveq1 6657	. . . . . . . . . 10 |- ( x = z -> ( x F y ) = ( z F y ) )
51	49, 50	opeq12d 4410	. . . . . . . . 9 |- ( x = z -> <. ( x + 1 ) , ( x F y ) >. = <. ( z + 1 ) , ( z F y ) >. )
52		oveq2 6658	. . . . . . . . . 10 |- ( y = w -> ( z F y ) = ( z F w ) )
53	52	opeq2d 4409	. . . . . . . . 9 |- ( y = w -> <. ( z + 1 ) , ( z F y ) >. = <. ( z + 1 ) , ( z F w ) >. )
54	51, 53	cbvmpt2v 6735	. . . . . . . 8 |- ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) = ( z e. _V , w e. _V |-> <. ( z + 1 ) , ( z F w ) >. )
55		opex 4932	. . . . . . . 8 |- <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. _V
56	46, 48, 54, 55	ovmpt2 6796	. . . . . . 7 |- ( ( ( G ` ( `' G ` B ) ) e. _V /\ ( 2nd ` ( R ` ( `' G ` B ) ) ) e. _V ) -> ( ( G ` ( `' G ` B ) ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
57	42, 43, 56	mp2an 708	. . . . . 6 |- ( ( G ` ( `' G ` B ) ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >.
58	41, 57	syl6eq 2672	. . . . 5 |- ( ( `' G ` B ) e. om -> ( R ` suc ( `' G ` B ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
59	58	fveq2d 6195	. . . 4 |- ( ( `' G ` B ) e. om -> ( 2nd ` ( R ` suc ( `' G ` B ) ) ) = ( 2nd ` <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. ) )
60		ovex 6678	. . . . 5 |- ( ( G ` ( `' G ` B ) ) + 1 ) e. _V
61		ovex 6678	. . . . 5 |- ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. _V
62	60, 61	op2nd 7177	. . . 4 |- ( 2nd ` <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) )
63	59, 62	syl6eq 2672	. . 3 |- ( ( `' G ` B ) e. om -> ( 2nd ` ( R ` suc ( `' G ` B ) ) ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
64	17, 63	syl 17	. 2 |- ( B e. ( ZZ>= ` C ) -> ( 2nd ` ( R ` suc ( `' G ` B ) ) ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
65	1, 2, 3, 4	uzrdglem 12756	. . . . . 6 |- ( B e. ( ZZ>= ` C ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
66	65, 5	syl6eleqr 2712	. . . . 5 |- ( B e. ( ZZ>= ` C ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. S )
67		funopfv 6235	. . . . 5 |- ( Fun S -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. S -> ( S ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
68	8, 66, 67	mpsyl 68	. . . 4 |- ( B e. ( ZZ>= ` C ) -> ( S ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) )
69	68	eqcomd 2628	. . 3 |- ( B e. ( ZZ>= ` C ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) = ( S ` B ) )
70	23, 69	oveq12d 6668	. 2 |- ( B e. ( ZZ>= ` C ) -> ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = ( B F ( S ` B ) ) )
71	31, 64, 70	3eqtrd 2660	1 |- ( B e. ( ZZ>= ` C ) -> ( S ` ( B + 1 ) ) = ( B F ( S ` B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   |` cres 5116   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   2ndc2nd 7167   reccrdg 7505   1c1 9937   + caddc 9939   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-2nd 7169  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:  seqp1  12816