Theorem icccmplem3 22627

Description: Lemma for icccmp 22628. (Contributed by Mario Carneiro, 13-Jun-2014.)

Hypotheses

Ref	Expression
icccmp.1	|- J = ( topGen ` ran (,) )
icccmp.2	|- T = ( J ↾t ( A [,] B ) )
icccmp.3	|- D = ( ( abs o. - ) |` ( RR X. RR ) )
icccmp.4	|- S = { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z }
icccmp.5	|- ( ph -> A e. RR )
icccmp.6	|- ( ph -> B e. RR )
icccmp.7	|- ( ph -> A <_ B )
icccmp.8	|- ( ph -> U C_ J )
icccmp.9	|- ( ph -> ( A [,] B ) C_ U. U )

Assertion

Ref	Expression
icccmplem3	|- ( ph -> B e. S )

Distinct variable groups:   x, z, B   x, A, z   x, D   x, T, z   z, J   x, U, z

Allowed substitution hints:   ph( x, z)   D( z)   S( x, z)   J( x)

Proof of Theorem icccmplem3

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

Step	Hyp	Ref	Expression
1		icccmp.9	. . . 4 |- ( ph -> ( A [,] B ) C_ U. U )
2		icccmp.4	. . . . . . . 8 |- S = { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z }
3		ssrab2 3687	. . . . . . . 8 |- { x e. ( A [,] B ) | E. z e. ( ~P U i^i Fin ) ( A [,] x ) C_ U. z } C_ ( A [,] B )
4	2, 3	eqsstri 3635	. . . . . . 7 |- S C_ ( A [,] B )
5		icccmp.5	. . . . . . . 8 |- ( ph -> A e. RR )
6		icccmp.6	. . . . . . . 8 |- ( ph -> B e. RR )
7		iccssre 12255	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
8	5, 6, 7	syl2anc 693	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ RR )
9	4, 8	syl5ss 3614	. . . . . 6 |- ( ph -> S C_ RR )
10		icccmp.1	. . . . . . . . 9 |- J = ( topGen ` ran (,) )
11		icccmp.2	. . . . . . . . 9 |- T = ( J ↾t ( A [,] B ) )
12		icccmp.3	. . . . . . . . 9 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
13		icccmp.7	. . . . . . . . 9 |- ( ph -> A <_ B )
14		icccmp.8	. . . . . . . . 9 |- ( ph -> U C_ J )
15	10, 11, 12, 2, 5, 6, 13, 14, 1	icccmplem1 22625	. . . . . . . 8 |- ( ph -> ( A e. S /\ A. y e. S y <_ B ) )
16	15	simpld 475	. . . . . . 7 |- ( ph -> A e. S )
17		ne0i 3921	. . . . . . 7 |- ( A e. S -> S =/= (/) )
18	16, 17	syl 17	. . . . . 6 |- ( ph -> S =/= (/) )
19	15	simprd 479	. . . . . . 7 |- ( ph -> A. y e. S y <_ B )
20		breq2 4657	. . . . . . . . 9 |- ( v = B -> ( y <_ v <-> y <_ B ) )
21	20	ralbidv 2986	. . . . . . . 8 |- ( v = B -> ( A. y e. S y <_ v <-> A. y e. S y <_ B ) )
22	21	rspcev 3309	. . . . . . 7 |- ( ( B e. RR /\ A. y e. S y <_ B ) -> E. v e. RR A. y e. S y <_ v )
23	6, 19, 22	syl2anc 693	. . . . . 6 |- ( ph -> E. v e. RR A. y e. S y <_ v )
24		suprcl 10983	. . . . . 6 |- ( ( S C_ RR /\ S =/= (/) /\ E. v e. RR A. y e. S y <_ v ) -> sup ( S , RR , < ) e. RR )
25	9, 18, 23, 24	syl3anc 1326	. . . . 5 |- ( ph -> sup ( S , RR , < ) e. RR )
26		suprub 10984	. . . . . 6 |- ( ( ( S C_ RR /\ S =/= (/) /\ E. v e. RR A. y e. S y <_ v ) /\ A e. S ) -> A <_ sup ( S , RR , < ) )
27	9, 18, 23, 16, 26	syl31anc 1329	. . . . 5 |- ( ph -> A <_ sup ( S , RR , < ) )
28		suprleub 10989	. . . . . . 7 |- ( ( ( S C_ RR /\ S =/= (/) /\ E. v e. RR A. y e. S y <_ v ) /\ B e. RR ) -> ( sup ( S , RR , < ) <_ B <-> A. y e. S y <_ B ) )
29	9, 18, 23, 6, 28	syl31anc 1329	. . . . . 6 |- ( ph -> ( sup ( S , RR , < ) <_ B <-> A. y e. S y <_ B ) )
30	19, 29	mpbird 247	. . . . 5 |- ( ph -> sup ( S , RR , < ) <_ B )
31		elicc2 12238	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( sup ( S , RR , < ) e. ( A [,] B ) <-> ( sup ( S , RR , < ) e. RR /\ A <_ sup ( S , RR , < ) /\ sup ( S , RR , < ) <_ B ) ) )
32	5, 6, 31	syl2anc 693	. . . . 5 |- ( ph -> ( sup ( S , RR , < ) e. ( A [,] B ) <-> ( sup ( S , RR , < ) e. RR /\ A <_ sup ( S , RR , < ) /\ sup ( S , RR , < ) <_ B ) ) )
33	25, 27, 30, 32	mpbir3and 1245	. . . 4 |- ( ph -> sup ( S , RR , < ) e. ( A [,] B ) )
34	1, 33	sseldd 3604	. . 3 |- ( ph -> sup ( S , RR , < ) e. U. U )
35		eluni2 4440	. . 3 |- ( sup ( S , RR , < ) e. U. U <-> E. u e. U sup ( S , RR , < ) e. u )
36	34, 35	sylib 208	. 2 |- ( ph -> E. u e. U sup ( S , RR , < ) e. u )
37	14	sselda 3603	. . . . 5 |- ( ( ph /\ u e. U ) -> u e. J )
38	12	rexmet 22594	. . . . . . 7 |- D e. ( *Met ` RR )
39		eqid 2622	. . . . . . . . . 10 |- ( MetOpen ` D ) = ( MetOpen ` D )
40	12, 39	tgioo 22599	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( MetOpen ` D )
41	10, 40	eqtri 2644	. . . . . . . 8 |- J = ( MetOpen ` D )
42	41	mopni2 22298	. . . . . . 7 |- ( ( D e. ( *Met ` RR ) /\ u e. J /\ sup ( S , RR , < ) e. u ) -> E. w e. RR+ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u )
43	38, 42	mp3an1 1411	. . . . . 6 |- ( ( u e. J /\ sup ( S , RR , < ) e. u ) -> E. w e. RR+ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u )
44	43	ex 450	. . . . 5 |- ( u e. J -> ( sup ( S , RR , < ) e. u -> E. w e. RR+ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) )
45	37, 44	syl 17	. . . 4 |- ( ( ph /\ u e. U ) -> ( sup ( S , RR , < ) e. u -> E. w e. RR+ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) )
46	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> A e. RR )
47	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> B e. RR )
48	13	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> A <_ B )
49	14	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> U C_ J )
50	1	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> ( A [,] B ) C_ U. U )
51		simplr 792	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> u e. U )
52		simprl 794	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> w e. RR+ )
53		simprr 796	. . . . . 6 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u )
54		eqid 2622	. . . . . 6 |- sup ( S , RR , < ) = sup ( S , RR , < )
55		eqid 2622	. . . . . 6 |- if ( ( sup ( S , RR , < ) + ( w / 2 ) ) <_ B , ( sup ( S , RR , < ) + ( w / 2 ) ) , B ) = if ( ( sup ( S , RR , < ) + ( w / 2 ) ) <_ B , ( sup ( S , RR , < ) + ( w / 2 ) ) , B )
56	10, 11, 12, 2, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55	icccmplem2 22626	. . . . 5 |- ( ( ( ph /\ u e. U ) /\ ( w e. RR+ /\ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u ) ) -> B e. S )
57	56	rexlimdvaa 3032	. . . 4 |- ( ( ph /\ u e. U ) -> ( E. w e. RR+ ( sup ( S , RR , < ) ( ball ` D ) w ) C_ u -> B e. S ) )
58	45, 57	syld 47	. . 3 |- ( ( ph /\ u e. U ) -> ( sup ( S , RR , < ) e. u -> B e. S ) )
59	58	rexlimdva 3031	. 2 |- ( ph -> ( E. u e. U sup ( S , RR , < ) e. u -> B e. S ) )
60	36, 59	mpd 15	1 |- ( ph -> B e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   RRcr 9935   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   abscabs 13974   ↾_t crest 16081   topGenctg 16098   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736

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  ax-pre-sup 10014

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-rmo 2920  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-int 4476  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  icccmp  22628