Theorem om2uzrdg 12755

Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F ( x , y ) and initial value A. Normally F is a function on the partition, and A is a member of the partition. See also comment in om2uz0i 12746. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

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 )

Assertion

Ref	Expression
om2uzrdg	|- ( B e. om -> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` 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)   G( x)

Proof of Theorem om2uzrdg

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( z = (/) -> ( R ` z ) = ( R ` (/) ) )
2		fveq2 6191	. . . 4 |- ( z = (/) -> ( G ` z ) = ( G ` (/) ) )
3	1	fveq2d 6195	. . . 4 |- ( z = (/) -> ( 2nd ` ( R ` z ) ) = ( 2nd ` ( R ` (/) ) ) )
4	2, 3	opeq12d 4410	. . 3 |- ( z = (/) -> <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. = <. ( G ` (/) ) , ( 2nd ` ( R ` (/) ) ) >. )
5	1, 4	eqeq12d 2637	. 2 |- ( z = (/) -> ( ( R ` z ) = <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. <-> ( R ` (/) ) = <. ( G ` (/) ) , ( 2nd ` ( R ` (/) ) ) >. ) )
6		fveq2 6191	. . 3 |- ( z = v -> ( R ` z ) = ( R ` v ) )
7		fveq2 6191	. . . 4 |- ( z = v -> ( G ` z ) = ( G ` v ) )
8	6	fveq2d 6195	. . . 4 |- ( z = v -> ( 2nd ` ( R ` z ) ) = ( 2nd ` ( R ` v ) ) )
9	7, 8	opeq12d 4410	. . 3 |- ( z = v -> <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. )
10	6, 9	eqeq12d 2637	. 2 |- ( z = v -> ( ( R ` z ) = <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. <-> ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) )
11		fveq2 6191	. . 3 |- ( z = suc v -> ( R ` z ) = ( R ` suc v ) )
12		fveq2 6191	. . . 4 |- ( z = suc v -> ( G ` z ) = ( G ` suc v ) )
13	11	fveq2d 6195	. . . 4 |- ( z = suc v -> ( 2nd ` ( R ` z ) ) = ( 2nd ` ( R ` suc v ) ) )
14	12, 13	opeq12d 4410	. . 3 |- ( z = suc v -> <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. = <. ( G ` suc v ) , ( 2nd ` ( R ` suc v ) ) >. )
15	11, 14	eqeq12d 2637	. 2 |- ( z = suc v -> ( ( R ` z ) = <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. <-> ( R ` suc v ) = <. ( G ` suc v ) , ( 2nd ` ( R ` suc v ) ) >. ) )
16		fveq2 6191	. . 3 |- ( z = B -> ( R ` z ) = ( R ` B ) )
17		fveq2 6191	. . . 4 |- ( z = B -> ( G ` z ) = ( G ` B ) )
18	16	fveq2d 6195	. . . 4 |- ( z = B -> ( 2nd ` ( R ` z ) ) = ( 2nd ` ( R ` B ) ) )
19	17, 18	opeq12d 4410	. . 3 |- ( z = B -> <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. )
20	16, 19	eqeq12d 2637	. 2 |- ( z = B -> ( ( R ` z ) = <. ( G ` z ) , ( 2nd ` ( R ` z ) ) >. <-> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. ) )
21		uzrdg.2	. . . . 5 |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )
22	21	fveq1i 6192	. . . 4 |- ( R ` (/) ) = ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` (/) )
23		opex 4932	. . . . 5 |- <. C , A >. e. _V
24		fr0g 7531	. . . . 5 |- ( <. C , A >. e. _V -> ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` (/) ) = <. C , A >. )
25	23, 24	ax-mp 5	. . . 4 |- ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` (/) ) = <. C , A >.
26	22, 25	eqtri 2644	. . 3 |- ( R ` (/) ) = <. C , A >.
27		om2uz.1	. . . . 5 |- C e. ZZ
28		om2uz.2	. . . . 5 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
29	27, 28	om2uz0i 12746	. . . 4 |- ( G ` (/) ) = C
30	26	fveq2i 6194	. . . . 5 |- ( 2nd ` ( R ` (/) ) ) = ( 2nd ` <. C , A >. )
31	27	elexi 3213	. . . . . 6 |- C e. _V
32		uzrdg.1	. . . . . 6 |- A e. _V
33	31, 32	op2nd 7177	. . . . 5 |- ( 2nd ` <. C , A >. ) = A
34	30, 33	eqtri 2644	. . . 4 |- ( 2nd ` ( R ` (/) ) ) = A
35	29, 34	opeq12i 4407	. . 3 |- <. ( G ` (/) ) , ( 2nd ` ( R ` (/) ) ) >. = <. C , A >.
36	26, 35	eqtr4i 2647	. 2 |- ( R ` (/) ) = <. ( G ` (/) ) , ( 2nd ` ( R ` (/) ) ) >.
37		frsuc 7532	. . . . . 6 |- ( v e. om -> ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` suc v ) = ( ( 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 ) ` v ) ) )
38	21	fveq1i 6192	. . . . . 6 |- ( R ` suc v ) = ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` suc v )
39	21	fveq1i 6192	. . . . . . 7 |- ( R ` v ) = ( ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om ) ` v )
40	39	fveq2i 6194	. . . . . 6 |- ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` v ) ) = ( ( 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 ) ` v ) )
41	37, 38, 40	3eqtr4g 2681	. . . . 5 |- ( v e. om -> ( R ` suc v ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` v ) ) )
42		fveq2 6191	. . . . . 6 |- ( ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. -> ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` v ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) )
43		df-ov 6653	. . . . . . 7 |- ( ( G ` v ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` v ) ) ) = ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. )
44		fvex 6201	. . . . . . . 8 |- ( G ` v ) e. _V
45		fvex 6201	. . . . . . . 8 |- ( 2nd ` ( R ` v ) ) e. _V
46		oveq1 6657	. . . . . . . . . 10 |- ( w = ( G ` v ) -> ( w + 1 ) = ( ( G ` v ) + 1 ) )
47		oveq1 6657	. . . . . . . . . 10 |- ( w = ( G ` v ) -> ( w F z ) = ( ( G ` v ) F z ) )
48	46, 47	opeq12d 4410	. . . . . . . . 9 |- ( w = ( G ` v ) -> <. ( w + 1 ) , ( w F z ) >. = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F z ) >. )
49		oveq2 6658	. . . . . . . . . 10 |- ( z = ( 2nd ` ( R ` v ) ) -> ( ( G ` v ) F z ) = ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) )
50	49	opeq2d 4409	. . . . . . . . 9 |- ( z = ( 2nd ` ( R ` v ) ) -> <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F z ) >. = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. )
51		oveq1 6657	. . . . . . . . . . 11 |- ( x = w -> ( x + 1 ) = ( w + 1 ) )
52		oveq1 6657	. . . . . . . . . . 11 |- ( x = w -> ( x F y ) = ( w F y ) )
53	51, 52	opeq12d 4410	. . . . . . . . . 10 |- ( x = w -> <. ( x + 1 ) , ( x F y ) >. = <. ( w + 1 ) , ( w F y ) >. )
54		oveq2 6658	. . . . . . . . . . 11 |- ( y = z -> ( w F y ) = ( w F z ) )
55	54	opeq2d 4409	. . . . . . . . . 10 |- ( y = z -> <. ( w + 1 ) , ( w F y ) >. = <. ( w + 1 ) , ( w F z ) >. )
56	53, 55	cbvmpt2v 6735	. . . . . . . . 9 |- ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) = ( w e. _V , z e. _V |-> <. ( w + 1 ) , ( w F z ) >. )
57		opex 4932	. . . . . . . . 9 |- <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. e. _V
58	48, 50, 56, 57	ovmpt2 6796	. . . . . . . 8 |- ( ( ( G ` v ) e. _V /\ ( 2nd ` ( R ` v ) ) e. _V ) -> ( ( G ` v ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` v ) ) ) = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. )
59	44, 45, 58	mp2an 708	. . . . . . 7 |- ( ( G ` v ) ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` v ) ) ) = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >.
60	43, 59	eqtr3i 2646	. . . . . 6 |- ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >.
61	42, 60	syl6eq 2672	. . . . 5 |- ( ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. -> ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` v ) ) = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. )
62	41, 61	sylan9eq 2676	. . . 4 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> ( R ` suc v ) = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. )
63	27, 28	om2uzsuci 12747	. . . . . 6 |- ( v e. om -> ( G ` suc v ) = ( ( G ` v ) + 1 ) )
64	63	adantr 481	. . . . 5 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> ( G ` suc v ) = ( ( G ` v ) + 1 ) )
65	62	fveq2d 6195	. . . . . 6 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> ( 2nd ` ( R ` suc v ) ) = ( 2nd ` <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. ) )
66		ovex 6678	. . . . . . 7 |- ( ( G ` v ) + 1 ) e. _V
67		ovex 6678	. . . . . . 7 |- ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) e. _V
68	66, 67	op2nd 7177	. . . . . 6 |- ( 2nd ` <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. ) = ( ( G ` v ) F ( 2nd ` ( R ` v ) ) )
69	65, 68	syl6eq 2672	. . . . 5 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> ( 2nd ` ( R ` suc v ) ) = ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) )
70	64, 69	opeq12d 4410	. . . 4 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> <. ( G ` suc v ) , ( 2nd ` ( R ` suc v ) ) >. = <. ( ( G ` v ) + 1 ) , ( ( G ` v ) F ( 2nd ` ( R ` v ) ) ) >. )
71	62, 70	eqtr4d 2659	. . 3 |- ( ( v e. om /\ ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. ) -> ( R ` suc v ) = <. ( G ` suc v ) , ( 2nd ` ( R ` suc v ) ) >. )
72	71	ex 450	. 2 |- ( v e. om -> ( ( R ` v ) = <. ( G ` v ) , ( 2nd ` ( R ` v ) ) >. -> ( R ` suc v ) = <. ( G ` suc v ) , ( 2nd ` ( R ` suc v ) ) >. ) )
73	5, 10, 15, 20, 36, 72	finds 7092	1 |- ( B e. om -> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   2ndc2nd 7167   reccrdg 7505   1c1 9937   + caddc 9939   ZZcz 11377

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

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  uzrdglem  12756  uzrdgfni  12757  uzrdgsuci  12759