Theorem heiborlem4 33613

Description: Lemma for heibor 33620. Using the function T constructed in heiborlem3 33612, construct an infinite path in G. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	|- J = ( MetOpen ` D )
heibor.3	|- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
heibor.4	|- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
heibor.5	|- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
heibor.6	|- ( ph -> D e. ( CMet ` X ) )
heibor.7	|- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
heibor.8	|- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
heibor.9	|- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
heibor.10	|- ( ph -> C G 0 )
heibor.11	|- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )

Assertion

Ref	Expression
heiborlem4	|- ( ( ph /\ A e. NN0 ) -> ( S ` A ) G A )

Distinct variable groups:   x, n, y, A   u, n, F, x, y   x, G   ph, x   m, n, u, v, x, y, z, D   T, m, n, x, y, z   B, n, u, v, y   m, J, n, u, v, x, y, z   U, n, u, v, x, y, z   S, m, n, u, v, x, y, z   m, X, n, u, v, x, y, z   C, m, n, u, v, y   n, K, x, y, z   x, B

Allowed substitution hints:   ph( y, z, v, u, m, n)   A( z, v, u, m)   B( z, m)   C( x, z)   T( v, u)   U( m)   F( z, v, m)   G( y, z, v, u, m, n)   K( v, u, m)

Proof of Theorem heiborlem4

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( x = 0 -> ( S ` x ) = ( S ` 0 ) )
2		id 22	. . . . 5 |- ( x = 0 -> x = 0 )
3	1, 2	breq12d 4666	. . . 4 |- ( x = 0 -> ( ( S ` x ) G x <-> ( S ` 0 ) G 0 ) )
4	3	imbi2d 330	. . 3 |- ( x = 0 -> ( ( ph -> ( S ` x ) G x ) <-> ( ph -> ( S ` 0 ) G 0 ) ) )
5		fveq2 6191	. . . . 5 |- ( x = k -> ( S ` x ) = ( S ` k ) )
6		id 22	. . . . 5 |- ( x = k -> x = k )
7	5, 6	breq12d 4666	. . . 4 |- ( x = k -> ( ( S ` x ) G x <-> ( S ` k ) G k ) )
8	7	imbi2d 330	. . 3 |- ( x = k -> ( ( ph -> ( S ` x ) G x ) <-> ( ph -> ( S ` k ) G k ) ) )
9		fveq2 6191	. . . . 5 |- ( x = ( k + 1 ) -> ( S ` x ) = ( S ` ( k + 1 ) ) )
10		id 22	. . . . 5 |- ( x = ( k + 1 ) -> x = ( k + 1 ) )
11	9, 10	breq12d 4666	. . . 4 |- ( x = ( k + 1 ) -> ( ( S ` x ) G x <-> ( S ` ( k + 1 ) ) G ( k + 1 ) ) )
12	11	imbi2d 330	. . 3 |- ( x = ( k + 1 ) -> ( ( ph -> ( S ` x ) G x ) <-> ( ph -> ( S ` ( k + 1 ) ) G ( k + 1 ) ) ) )
13		fveq2 6191	. . . . 5 |- ( x = A -> ( S ` x ) = ( S ` A ) )
14		id 22	. . . . 5 |- ( x = A -> x = A )
15	13, 14	breq12d 4666	. . . 4 |- ( x = A -> ( ( S ` x ) G x <-> ( S ` A ) G A ) )
16	15	imbi2d 330	. . 3 |- ( x = A -> ( ( ph -> ( S ` x ) G x ) <-> ( ph -> ( S ` A ) G A ) ) )
17		heibor.11	. . . . . . 7 |- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )
18	17	fveq1i 6192	. . . . . 6 |- ( S ` 0 ) = ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` 0 )
19		0z 11388	. . . . . . 7 |- 0 e. ZZ
20		seq1 12814	. . . . . . 7 |- ( 0 e. ZZ -> ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` 0 ) = ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` 0 ) )
21	19, 20	ax-mp 5	. . . . . 6 |- ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` 0 ) = ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` 0 )
22	18, 21	eqtri 2644	. . . . 5 |- ( S ` 0 ) = ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` 0 )
23		0nn0 11307	. . . . . 6 |- 0 e. NN0
24		heibor.10	. . . . . . 7 |- ( ph -> C G 0 )
25		heibor.4	. . . . . . . . 9 |- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
26	25	relopabi 5245	. . . . . . . 8 |- Rel G
27	26	brrelexi 5158	. . . . . . 7 |- ( C G 0 -> C e. _V )
28	24, 27	syl 17	. . . . . 6 |- ( ph -> C e. _V )
29		iftrue 4092	. . . . . . 7 |- ( m = 0 -> if ( m = 0 , C , ( m - 1 ) ) = C )
30		eqid 2622	. . . . . . 7 |- ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) = ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) )
31	29, 30	fvmptg 6280	. . . . . 6 |- ( ( 0 e. NN0 /\ C e. _V ) -> ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` 0 ) = C )
32	23, 28, 31	sylancr 695	. . . . 5 |- ( ph -> ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` 0 ) = C )
33	22, 32	syl5eq 2668	. . . 4 |- ( ph -> ( S ` 0 ) = C )
34	33, 24	eqbrtrd 4675	. . 3 |- ( ph -> ( S ` 0 ) G 0 )
35		df-br 4654	. . . . . 6 |- ( ( S ` k ) G k <-> <. ( S ` k ) , k >. e. G )
36		heibor.9	. . . . . . 7 |- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
37		fveq2 6191	. . . . . . . . . . 11 |- ( x = <. ( S ` k ) , k >. -> ( T ` x ) = ( T ` <. ( S ` k ) , k >. ) )
38		df-ov 6653	. . . . . . . . . . 11 |- ( ( S ` k ) T k ) = ( T ` <. ( S ` k ) , k >. )
39	37, 38	syl6eqr 2674	. . . . . . . . . 10 |- ( x = <. ( S ` k ) , k >. -> ( T ` x ) = ( ( S ` k ) T k ) )
40		fvex 6201	. . . . . . . . . . . 12 |- ( S ` k ) e. _V
41		vex 3203	. . . . . . . . . . . 12 |- k e. _V
42	40, 41	op2ndd 7179	. . . . . . . . . . 11 |- ( x = <. ( S ` k ) , k >. -> ( 2nd ` x ) = k )
43	42	oveq1d 6665	. . . . . . . . . 10 |- ( x = <. ( S ` k ) , k >. -> ( ( 2nd ` x ) + 1 ) = ( k + 1 ) )
44	39, 43	breq12d 4666	. . . . . . . . 9 |- ( x = <. ( S ` k ) , k >. -> ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) <-> ( ( S ` k ) T k ) G ( k + 1 ) ) )
45		fveq2 6191	. . . . . . . . . . . 12 |- ( x = <. ( S ` k ) , k >. -> ( B ` x ) = ( B ` <. ( S ` k ) , k >. ) )
46		df-ov 6653	. . . . . . . . . . . 12 |- ( ( S ` k ) B k ) = ( B ` <. ( S ` k ) , k >. )
47	45, 46	syl6eqr 2674	. . . . . . . . . . 11 |- ( x = <. ( S ` k ) , k >. -> ( B ` x ) = ( ( S ` k ) B k ) )
48	39, 43	oveq12d 6668	. . . . . . . . . . 11 |- ( x = <. ( S ` k ) , k >. -> ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) = ( ( ( S ` k ) T k ) B ( k + 1 ) ) )
49	47, 48	ineq12d 3815	. . . . . . . . . 10 |- ( x = <. ( S ` k ) , k >. -> ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) = ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) )
50	49	eleq1d 2686	. . . . . . . . 9 |- ( x = <. ( S ` k ) , k >. -> ( ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K <-> ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) )
51	44, 50	anbi12d 747	. . . . . . . 8 |- ( x = <. ( S ` k ) , k >. -> ( ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) <-> ( ( ( S ` k ) T k ) G ( k + 1 ) /\ ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) ) )
52	51	rspccv 3306	. . . . . . 7 |- ( A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) -> ( <. ( S ` k ) , k >. e. G -> ( ( ( S ` k ) T k ) G ( k + 1 ) /\ ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) ) )
53	36, 52	syl 17	. . . . . 6 |- ( ph -> ( <. ( S ` k ) , k >. e. G -> ( ( ( S ` k ) T k ) G ( k + 1 ) /\ ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) ) )
54	35, 53	syl5bi 232	. . . . 5 |- ( ph -> ( ( S ` k ) G k -> ( ( ( S ` k ) T k ) G ( k + 1 ) /\ ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) ) )
55		seqp1 12816	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` 0 ) -> ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` ( k + 1 ) ) = ( ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) ) )
56		nn0uz 11722	. . . . . . . . . . 11 |- NN0 = ( ZZ>= ` 0 )
57	55, 56	eleq2s 2719	. . . . . . . . . 10 |- ( k e. NN0 -> ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` ( k + 1 ) ) = ( ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) ) )
58	17	fveq1i 6192	. . . . . . . . . 10 |- ( S ` ( k + 1 ) ) = ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` ( k + 1 ) )
59	17	fveq1i 6192	. . . . . . . . . . 11 |- ( S ` k ) = ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` k )
60	59	oveq1i 6660	. . . . . . . . . 10 |- ( ( S ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) ) = ( ( seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ) ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) )
61	57, 58, 60	3eqtr4g 2681	. . . . . . . . 9 |- ( k e. NN0 -> ( S ` ( k + 1 ) ) = ( ( S ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) ) )
62		peano2nn0 11333	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
63		nn0p1nn 11332	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> ( k + 1 ) e. NN )
64		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( ( k + 1 ) e. NN -> ( k + 1 ) =/= 0 )
65	64	neneqd 2799	. . . . . . . . . . . . . 14 |- ( ( k + 1 ) e. NN -> -. ( k + 1 ) = 0 )
66		iffalse 4095	. . . . . . . . . . . . . 14 |- ( -. ( k + 1 ) = 0 -> if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) = ( ( k + 1 ) - 1 ) )
67	63, 65, 66	3syl 18	. . . . . . . . . . . . 13 |- ( k e. NN0 -> if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) = ( ( k + 1 ) - 1 ) )
68		ovex 6678	. . . . . . . . . . . . 13 |- ( ( k + 1 ) - 1 ) e. _V
69	67, 68	syl6eqel 2709	. . . . . . . . . . . 12 |- ( k e. NN0 -> if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) e. _V )
70		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( m = ( k + 1 ) -> ( m = 0 <-> ( k + 1 ) = 0 ) )
71		oveq1 6657	. . . . . . . . . . . . . 14 |- ( m = ( k + 1 ) -> ( m - 1 ) = ( ( k + 1 ) - 1 ) )
72	70, 71	ifbieq2d 4111	. . . . . . . . . . . . 13 |- ( m = ( k + 1 ) -> if ( m = 0 , C , ( m - 1 ) ) = if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) )
73	72, 30	fvmptg 6280	. . . . . . . . . . . 12 |- ( ( ( k + 1 ) e. NN0 /\ if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) e. _V ) -> ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) = if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) )
74	62, 69, 73	syl2anc 693	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) = if ( ( k + 1 ) = 0 , C , ( ( k + 1 ) - 1 ) ) )
75		nn0cn 11302	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
76		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
77		pncan 10287	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC ) -> ( ( k + 1 ) - 1 ) = k )
78	75, 76, 77	sylancl 694	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( k + 1 ) - 1 ) = k )
79	74, 67, 78	3eqtrd 2660	. . . . . . . . . 10 |- ( k e. NN0 -> ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) = k )
80	79	oveq2d 6666	. . . . . . . . 9 |- ( k e. NN0 -> ( ( S ` k ) T ( ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) ` ( k + 1 ) ) ) = ( ( S ` k ) T k ) )
81	61, 80	eqtrd 2656	. . . . . . . 8 |- ( k e. NN0 -> ( S ` ( k + 1 ) ) = ( ( S ` k ) T k ) )
82	81	breq1d 4663	. . . . . . 7 |- ( k e. NN0 -> ( ( S ` ( k + 1 ) ) G ( k + 1 ) <-> ( ( S ` k ) T k ) G ( k + 1 ) ) )
83	82	biimprd 238	. . . . . 6 |- ( k e. NN0 -> ( ( ( S ` k ) T k ) G ( k + 1 ) -> ( S ` ( k + 1 ) ) G ( k + 1 ) ) )
84	83	adantrd 484	. . . . 5 |- ( k e. NN0 -> ( ( ( ( S ` k ) T k ) G ( k + 1 ) /\ ( ( ( S ` k ) B k ) i^i ( ( ( S ` k ) T k ) B ( k + 1 ) ) ) e. K ) -> ( S ` ( k + 1 ) ) G ( k + 1 ) ) )
85	54, 84	syl9r 78	. . . 4 |- ( k e. NN0 -> ( ph -> ( ( S ` k ) G k -> ( S ` ( k + 1 ) ) G ( k + 1 ) ) ) )
86	85	a2d 29	. . 3 |- ( k e. NN0 -> ( ( ph -> ( S ` k ) G k ) -> ( ph -> ( S ` ( k + 1 ) ) G ( k + 1 ) ) ) )
87	4, 8, 12, 16, 34, 86	nn0ind 11472	. 2 |- ( A e. NN0 -> ( ph -> ( S ` A ) G A ) )
88	87	impcom 446	1 |- ( ( ph /\ A e. NN0 ) -> ( S ` A ) G A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ^cexp 12860   ballcbl 19733   MetOpencmopn 19736   CMetcms 23052

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  df-seq 12802

This theorem is referenced by:  heiborlem5  33614  heiborlem6  33615  heiborlem8  33617