Theorem frecuzrdgsuc 10414

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10399 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 28-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
frecuzrdgrrn.a	|- ( ph -> A e. S )
frecuzrdgrrn.f	|- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
frecuzrdgrrn.2	|- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
frecuzrdgtcl.3	|- ( ph -> T = ran R )

Assertion

Ref	Expression
frecuzrdgsuc	|- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( B F ( T ` B ) ) )

Distinct variable groups:   y, A   x, C, y   y, G   x, F, y   x, S, y   ph, x, y   x, B, y

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

Proof of Theorem frecuzrdgsuc

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

Step	Hyp	Ref	Expression
1		frec2uz.1	. . . . . . 7 |- ( ph -> C e. ZZ )
2	1	adantr 276	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> C e. ZZ )
3		frec2uz.2	. . . . . 6 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
4		frecuzrdgrrn.a	. . . . . . 7 |- ( ph -> A e. S )
5	4	adantr 276	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> A e. S )
6		frecuzrdgrrn.f	. . . . . . 7 |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
7	6	adantlr 477	. . . . . 6 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )
8		frecuzrdgrrn.2	. . . . . 6 |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )
9		peano2uz 9583	. . . . . . 7 |- ( B e. ( ZZ>= ` C ) -> ( B + 1 ) e. ( ZZ>= ` C ) )
10	9	adantl 277	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( B + 1 ) e. ( ZZ>= ` C ) )
11	2, 3, 5, 7, 8, 10	frecuzrdglem 10411	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. ran R )
12		frecuzrdgtcl.3	. . . . . 6 |- ( ph -> T = ran R )
13	12	adantr 276	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> T = ran R )
14	11, 13	eleqtrrd 2257	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. T )
15	1, 3, 4, 6, 8, 12	frecuzrdgtcl 10412	. . . . . . 7 |- ( ph -> T : ( ZZ>= ` C ) --> S )
16		ffun 5369	. . . . . . 7 |- ( T : ( ZZ>= ` C ) --> S -> Fun T )
17	15, 16	syl 14	. . . . . 6 |- ( ph -> Fun T )
18		funopfv 5556	. . . . . 6 |- ( Fun T -> ( <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. T -> ( T ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) ) )
19	17, 18	syl 14	. . . . 5 |- ( ph -> ( <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. T -> ( T ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) ) )
20	19	adantr 276	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( <. ( B + 1 ) , ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) >. e. T -> ( T ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) ) )
21	14, 20	mpd 13	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) )
22	1, 3	frec2uzf1od 10406	. . . . . . . . 9 |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
23		f1ocnvdm 5782	. . . . . . . . 9 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
24	22, 23	sylan 283	. . . . . . . 8 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` B ) e. om )
25	2, 3, 24	frec2uzsucd 10401	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( G ` suc ( `' G ` B ) ) = ( ( G ` ( `' G ` B ) ) + 1 ) )
26		f1ocnvfv2 5779	. . . . . . . . 9 |- ( ( G : om -1-1-onto-> ( ZZ>= ` C ) /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) = B )
27	22, 26	sylan 283	. . . . . . . 8 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) = B )
28	27	oveq1d 5890	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` ( `' G ` B ) ) + 1 ) = ( B + 1 ) )
29	25, 28	eqtrd 2210	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( G ` suc ( `' G ` B ) ) = ( B + 1 ) )
30		peano2 4595	. . . . . . . 8 |- ( ( `' G ` B ) e. om -> suc ( `' G ` B ) e. om )
31	24, 30	syl 14	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> suc ( `' G ` B ) e. om )
32		f1ocnvfv 5780	. . . . . . 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 ) ) )
33	22, 31, 32	syl2an2r 595	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` suc ( `' G ` B ) ) = ( B + 1 ) -> ( `' G ` ( B + 1 ) ) = suc ( `' G ` B ) ) )
34	29, 33	mpd 13	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( `' G ` ( B + 1 ) ) = suc ( `' G ` B ) )
35	34	fveq2d 5520	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` ( B + 1 ) ) ) = ( R ` suc ( `' G ` B ) ) )
36	35	fveq2d 5520	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` ( B + 1 ) ) ) ) = ( 2nd ` ( R ` suc ( `' G ` B ) ) ) )
37	21, 36	eqtrd 2210	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( 2nd ` ( R ` suc ( `' G ` B ) ) ) )
38		1st2nd2 6176	. . . . . . . . . . 11 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
39	38	adantl 277	. . . . . . . . . 10 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
40	39	fveq2d 5520	. . . . . . . . 9 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
41		df-ov 5878	. . . . . . . . . . 11 |- ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
42		xp1st 6166	. . . . . . . . . . . . 13 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
43	42	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 1st ` z ) e. ( ZZ>= ` C ) )
44		xp2nd 6167	. . . . . . . . . . . . 13 |- ( z e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` z ) e. S )
45	44	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( 2nd ` z ) e. S )
46		peano2uz 9583	. . . . . . . . . . . . . 14 |- ( ( 1st ` z ) e. ( ZZ>= ` C ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
47	43, 46	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) )
48		oveq2 5883	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) F y ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
49	48	eleq1d 2246	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) F y ) e. S <-> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
50		oveq1 5882	. . . . . . . . . . . . . . . . 17 |- ( x = ( 1st ` z ) -> ( x F y ) = ( ( 1st ` z ) F y ) )
51	50	eleq1d 2246	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` z ) -> ( ( x F y ) e. S <-> ( ( 1st ` z ) F y ) e. S ) )
52	51	ralbidv 2477	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` z ) -> ( A. y e. S ( x F y ) e. S <-> A. y e. S ( ( 1st ` z ) F y ) e. S ) )
53	6	ralrimivva 2559	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
54	53	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. x e. ( ZZ>= ` C ) A. y e. S ( x F y ) e. S )
55	52, 54, 43	rspcdva 2847	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> A. y e. S ( ( 1st ` z ) F y ) e. S )
56	49, 55, 45	rspcdva 2847	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) e. S )
57		opelxp 4657	. . . . . . . . . . . . 13 |- ( <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) <-> ( ( ( 1st ` z ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( 1st ` z ) F ( 2nd ` z ) ) e. S ) )
58	47, 56, 57	sylanbrc 417	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
59		oveq1 5882	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` z ) -> ( x + 1 ) = ( ( 1st ` z ) + 1 ) )
60	59, 50	opeq12d 3787	. . . . . . . . . . . . 13 |- ( x = ( 1st ` z ) -> <. ( x + 1 ) , ( x F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. )
61	48	opeq2d 3786	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` z ) -> <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F y ) >. = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
62		eqid 2177	. . . . . . . . . . . . 13 |- ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) = ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. )
63	60, 61, 62	ovmpog 6009	. . . . . . . . . . . 12 |- ( ( ( 1st ` z ) e. ( ZZ>= ` C ) /\ ( 2nd ` z ) e. S /\ <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
64	43, 45, 58, 63	syl3anc 1238	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( 1st ` z ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` z ) ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
65	41, 64	eqtr3id 2224	. . . . . . . . . 10 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = <. ( ( 1st ` z ) + 1 ) , ( ( 1st ` z ) F ( 2nd ` z ) ) >. )
66	65, 58	eqeltrd 2254	. . . . . . . . 9 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) e. ( ( ZZ>= ` C ) X. S ) )
67	40, 66	eqeltrd 2254	. . . . . . . 8 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ z e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
68	67	ralrimiva 2550	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> A. z e. ( ( ZZ>= ` C ) X. S ) ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) )
69		uzid 9542	. . . . . . . . 9 |- ( C e. ZZ -> C e. ( ZZ>= ` C ) )
70	2, 69	syl 14	. . . . . . . 8 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> C e. ( ZZ>= ` C ) )
71		opelxp 4657	. . . . . . . 8 |- ( <. C , A >. e. ( ( ZZ>= ` C ) X. S ) <-> ( C e. ( ZZ>= ` C ) /\ A e. S ) )
72	70, 5, 71	sylanbrc 417	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. C , A >. e. ( ( ZZ>= ` C ) X. S ) )
73		frecsuc 6408	. . . . . . 7 |- ( ( A. z e. ( ( ZZ>= ` C ) X. S ) ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` z ) e. ( ( ZZ>= ` C ) X. S ) /\ <. C , A >. e. ( ( ZZ>= ` C ) X. S ) /\ ( `' G ` B ) e. om ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` ( `' G ` B ) ) ) )
74	68, 72, 24, 73	syl3anc 1238	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` ( `' G ` B ) ) ) )
75	8	fveq1i 5517	. . . . . 6 |- ( R ` suc ( `' G ` B ) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` suc ( `' G ` B ) )
76	8	fveq1i 5517	. . . . . . 7 |- ( R ` ( `' G ` B ) ) = (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` ( `' G ` B ) )
77	76	fveq2i 5519	. . . . . 6 |- ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` (frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) ` ( `' G ` B ) ) )
78	74, 75, 77	3eqtr4g 2235	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) )
79	2, 3, 5, 7, 8, 24	frec2uzrdg 10409	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) = <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
80	79	fveq2d 5520	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. ) )
81		df-ov 5878	. . . . . 6 |- ( ( G ` ( `' G ` B ) ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` <. ( G ` ( `' G ` B ) ) , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. )
82	80, 81	eqtr4di 2228	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ` ( R ` ( `' G ` B ) ) ) = ( ( G ` ( `' G ` B ) ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
83	2, 3, 24	frec2uzuzd 10402	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( G ` ( `' G ` B ) ) e. ( ZZ>= ` C ) )
84	2, 3, 5, 7, 8	frecuzrdgrrn 10408	. . . . . . . 8 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ ( `' G ` B ) e. om ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
85	24, 84	mpdan 421	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) )
86		xp2nd 6167	. . . . . . 7 |- ( ( R ` ( `' G ` B ) ) e. ( ( ZZ>= ` C ) X. S ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
87	85, 86	syl 14	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S )
88	28, 10	eqeltrd 2254	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` ( `' G ` B ) ) + 1 ) e. ( ZZ>= ` C ) )
89	7	caovclg 6027	. . . . . . . 8 |- ( ( ( ph /\ B e. ( ZZ>= ` C ) ) /\ ( z e. ( ZZ>= ` C ) /\ w e. S ) ) -> ( z F w ) e. S )
90	89, 83, 87	caovcld 6028	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. S )
91		opelxp 4657	. . . . . . 7 |- ( <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) <-> ( ( ( G ` ( `' G ` B ) ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. S ) )
92	88, 90, 91	sylanbrc 417	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) )
93		oveq1 5882	. . . . . . . 8 |- ( z = ( G ` ( `' G ` B ) ) -> ( z + 1 ) = ( ( G ` ( `' G ` B ) ) + 1 ) )
94		oveq1 5882	. . . . . . . 8 |- ( z = ( G ` ( `' G ` B ) ) -> ( z F w ) = ( ( G ` ( `' G ` B ) ) F w ) )
95	93, 94	opeq12d 3787	. . . . . . 7 |- ( z = ( G ` ( `' G ` B ) ) -> <. ( z + 1 ) , ( z F w ) >. = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F w ) >. )
96		oveq2 5883	. . . . . . . 8 |- ( w = ( 2nd ` ( R ` ( `' G ` B ) ) ) -> ( ( G ` ( `' G ` B ) ) F w ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
97	96	opeq2d 3786	. . . . . . 7 |- ( 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 ) ) ) ) >. )
98		oveq1 5882	. . . . . . . . 9 |- ( x = z -> ( x + 1 ) = ( z + 1 ) )
99		oveq1 5882	. . . . . . . . 9 |- ( x = z -> ( x F y ) = ( z F y ) )
100	98, 99	opeq12d 3787	. . . . . . . 8 |- ( x = z -> <. ( x + 1 ) , ( x F y ) >. = <. ( z + 1 ) , ( z F y ) >. )
101		oveq2 5883	. . . . . . . . 9 |- ( y = w -> ( z F y ) = ( z F w ) )
102	101	opeq2d 3786	. . . . . . . 8 |- ( y = w -> <. ( z + 1 ) , ( z F y ) >. = <. ( z + 1 ) , ( z F w ) >. )
103	100, 102	cbvmpov 5955	. . . . . . 7 |- ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) = ( z e. ( ZZ>= ` C ) , w e. S |-> <. ( z + 1 ) , ( z F w ) >. )
104	95, 97, 103	ovmpog 6009	. . . . . 6 |- ( ( ( G ` ( `' G ` B ) ) e. ( ZZ>= ` C ) /\ ( 2nd ` ( R ` ( `' G ` B ) ) ) e. S /\ <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. e. ( ( ZZ>= ` C ) X. S ) ) -> ( ( G ` ( `' G ` B ) ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
105	83, 87, 92, 104	syl3anc 1238	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` ( `' G ` B ) ) ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
106	78, 82, 105	3eqtrd 2214	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( R ` suc ( `' G ` B ) ) = <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. )
107	106	fveq2d 5520	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` suc ( `' G ` B ) ) ) = ( 2nd ` <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. ) )
108		op2ndg 6152	. . . 4 |- ( ( ( ( G ` ( `' G ` B ) ) + 1 ) e. ( ZZ>= ` C ) /\ ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) e. S ) -> ( 2nd ` <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
109	88, 90, 108	syl2anc 411	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` <. ( ( G ` ( `' G ` B ) ) + 1 ) , ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) >. ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
110	107, 109	eqtrd 2210	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` suc ( `' G ` B ) ) ) = ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
111		simpr 110	. . . . . . 7 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> B e. ( ZZ>= ` C ) )
112	2, 3, 5, 7, 8, 111	frecuzrdglem 10411	. . . . . 6 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
113	112, 13	eleqtrrd 2257	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T )
114		funopfv 5556	. . . . . . 7 |- ( Fun T -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
115	17, 114	syl 14	. . . . . 6 |- ( ph -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
116	115	adantr 276	. . . . 5 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. T -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) ) )
117	113, 116	mpd 13	. . . 4 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` B ) = ( 2nd ` ( R ` ( `' G ` B ) ) ) )
118	117	eqcomd 2183	. . 3 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( 2nd ` ( R ` ( `' G ` B ) ) ) = ( T ` B ) )
119	27, 118	oveq12d 5893	. 2 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( ( G ` ( `' G ` B ) ) F ( 2nd ` ( R ` ( `' G ` B ) ) ) ) = ( B F ( T ` B ) ) )
120	37, 110, 119	3eqtrd 2214	1 |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( B F ( T ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   <.cop 3596   |-> cmpt 4065   suc csuc 4366   omcom 4590   X. cxp 4625   `'ccnv 4626   ran crn 4628   Fun wfun 5211   -->wf 5213   -1-1-onto->wf1o 5216   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   1stc1st 6139   2ndc2nd 6140  freccfrec 6391   1c1 7812   + caddc 7814   ZZcz 9253   ZZ>=cuz 9528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529

This theorem is referenced by: (None)