Theorem pnfneige0 29997

Description: A neighborhood of +oo contains an unbounded interval based at a real number. See pnfnei 21024. (Contributed by Thierry Arnoux, 31-Jul-2017.)

Hypothesis

Ref	Expression
pnfneige0.j	|- J = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )

Assertion

Ref	Expression
pnfneige0	|- ( ( A e. J /\ +oo e. A ) -> E. x e. RR ( x (,] +oo ) C_ A )

Distinct variable group:   x, A

Allowed substitution hint:   J( x)

Proof of Theorem pnfneige0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0red 10041	. . . 4 |- ( ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) /\ y < 0 ) -> 0 e. RR )
2		simpllr 799	. . . 4 |- ( ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) /\ -. y < 0 ) -> y e. RR )
3	1, 2	ifclda 4120	. . 3 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> if ( y < 0 , 0 , y ) e. RR )
4		rexr 10085	. . . . . . 7 |- ( y e. RR -> y e. RR* )
5		0xr 10086	. . . . . . . 8 |- 0 e. RR*
6	5	a1i 11	. . . . . . 7 |- ( y e. RR -> 0 e. RR* )
7		pnfxr 10092	. . . . . . . 8 |- +oo e. RR*
8	7	a1i 11	. . . . . . 7 |- ( y e. RR -> +oo e. RR* )
9		iocinif 29543	. . . . . . 7 |- ( ( y e. RR* /\ 0 e. RR* /\ +oo e. RR* ) -> ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) = if ( y < 0 , ( 0 (,] +oo ) , ( y (,] +oo ) ) )
10	4, 6, 8, 9	syl3anc 1326	. . . . . 6 |- ( y e. RR -> ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) = if ( y < 0 , ( 0 (,] +oo ) , ( y (,] +oo ) ) )
11		ovif 6737	. . . . . 6 |- ( if ( y < 0 , 0 , y ) (,] +oo ) = if ( y < 0 , ( 0 (,] +oo ) , ( y (,] +oo ) )
12	10, 11	syl6reqr 2675	. . . . 5 |- ( y e. RR -> ( if ( y < 0 , 0 , y ) (,] +oo ) = ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) )
13	12	ad2antlr 763	. . . 4 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( if ( y < 0 , 0 , y ) (,] +oo ) = ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) )
14		iocssicc 12261	. . . . . 6 |- ( 0 (,] +oo ) C_ ( 0 [,] +oo )
15		sslin 3839	. . . . . 6 |- ( ( 0 (,] +oo ) C_ ( 0 [,] +oo ) -> ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) C_ ( ( y (,] +oo ) i^i ( 0 [,] +oo ) ) )
16	14, 15	mp1i 13	. . . . 5 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) C_ ( ( y (,] +oo ) i^i ( 0 [,] +oo ) ) )
17		simpr 477	. . . . . 6 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) )
18		ssin 3835	. . . . . . . 8 |- ( ( ( y (,] +oo ) C_ A /\ ( y (,] +oo ) C_ ( 0 (,] +oo ) ) <-> ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) )
19	18	biimpri 218	. . . . . . 7 |- ( ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) -> ( ( y (,] +oo ) C_ A /\ ( y (,] +oo ) C_ ( 0 (,] +oo ) ) )
20	19	simpld 475	. . . . . 6 |- ( ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) -> ( y (,] +oo ) C_ A )
21		ssinss1 3841	. . . . . 6 |- ( ( y (,] +oo ) C_ A -> ( ( y (,] +oo ) i^i ( 0 [,] +oo ) ) C_ A )
22	17, 20, 21	3syl 18	. . . . 5 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( ( y (,] +oo ) i^i ( 0 [,] +oo ) ) C_ A )
23	16, 22	sstrd 3613	. . . 4 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( ( y (,] +oo ) i^i ( 0 (,] +oo ) ) C_ A )
24	13, 23	eqsstrd 3639	. . 3 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> ( if ( y < 0 , 0 , y ) (,] +oo ) C_ A )
25		oveq1 6657	. . . . 5 |- ( x = if ( y < 0 , 0 , y ) -> ( x (,] +oo ) = ( if ( y < 0 , 0 , y ) (,] +oo ) )
26	25	sseq1d 3632	. . . 4 |- ( x = if ( y < 0 , 0 , y ) -> ( ( x (,] +oo ) C_ A <-> ( if ( y < 0 , 0 , y ) (,] +oo ) C_ A ) )
27	26	rspcev 3309	. . 3 |- ( ( if ( y < 0 , 0 , y ) e. RR /\ ( if ( y < 0 , 0 , y ) (,] +oo ) C_ A ) -> E. x e. RR ( x (,] +oo ) C_ A )
28	3, 24, 27	syl2anc 693	. 2 |- ( ( ( ( A e. J /\ +oo e. A ) /\ y e. RR ) /\ ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) ) -> E. x e. RR ( x (,] +oo ) C_ A )
29		letopon 21009	. . . . . . . . . 10 |- (ordTop ` <_ ) e. (TopOn ` RR* )
30		iccssxr 12256	. . . . . . . . . 10 |- ( 0 [,] +oo ) C_ RR*
31		resttopon 20965	. . . . . . . . . 10 |- ( ( (ordTop ` <_ ) e. (TopOn ` RR* ) /\ ( 0 [,] +oo ) C_ RR* ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) ) )
32	29, 30, 31	mp2an 708	. . . . . . . . 9 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) )
33	32	topontopi 20720	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Top
34	33	a1i 11	. . . . . . 7 |- ( A e. J -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Top )
35		ovex 6678	. . . . . . . 8 |- ( 0 (,] +oo ) e. _V
36	35	a1i 11	. . . . . . 7 |- ( A e. J -> ( 0 (,] +oo ) e. _V )
37		pnfneige0.j	. . . . . . . . . 10 |- J = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )
38		xrge0topn 29989	. . . . . . . . . 10 |- ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
39	37, 38	eqtri 2644	. . . . . . . . 9 |- J = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
40	39	eleq2i 2693	. . . . . . . 8 |- ( A e. J <-> A e. ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
41	40	biimpi 206	. . . . . . 7 |- ( A e. J -> A e. ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
42		elrestr 16089	. . . . . . 7 |- ( ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Top /\ ( 0 (,] +oo ) e. _V /\ A e. ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) -> ( A i^i ( 0 (,] +oo ) ) e. ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ↾t ( 0 (,] +oo ) ) )
43	34, 36, 41, 42	syl3anc 1326	. . . . . 6 |- ( A e. J -> ( A i^i ( 0 (,] +oo ) ) e. ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ↾t ( 0 (,] +oo ) ) )
44		letop 21010	. . . . . . 7 |- (ordTop ` <_ ) e. Top
45		ovex 6678	. . . . . . 7 |- ( 0 [,] +oo ) e. _V
46		restabs 20969	. . . . . . 7 |- ( ( (ordTop ` <_ ) e. Top /\ ( 0 (,] +oo ) C_ ( 0 [,] +oo ) /\ ( 0 [,] +oo ) e. _V ) -> ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ↾t ( 0 (,] +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 (,] +oo ) ) )
47	44, 14, 45, 46	mp3an 1424	. . . . . 6 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ↾t ( 0 (,] +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 (,] +oo ) )
48	43, 47	syl6eleq 2711	. . . . 5 |- ( A e. J -> ( A i^i ( 0 (,] +oo ) ) e. ( (ordTop ` <_ ) ↾t ( 0 (,] +oo ) ) )
49	44	a1i 11	. . . . . 6 |- ( A e. J -> (ordTop ` <_ ) e. Top )
50		iocpnfordt 21019	. . . . . . 7 |- ( 0 (,] +oo ) e. (ordTop ` <_ )
51	50	a1i 11	. . . . . 6 |- ( A e. J -> ( 0 (,] +oo ) e. (ordTop ` <_ ) )
52		ssid 3624	. . . . . . 7 |- ( 0 (,] +oo ) C_ ( 0 (,] +oo )
53	52	a1i 11	. . . . . 6 |- ( A e. J -> ( 0 (,] +oo ) C_ ( 0 (,] +oo ) )
54		inss2 3834	. . . . . . 7 |- ( A i^i ( 0 (,] +oo ) ) C_ ( 0 (,] +oo )
55	54	a1i 11	. . . . . 6 |- ( A e. J -> ( A i^i ( 0 (,] +oo ) ) C_ ( 0 (,] +oo ) )
56		restopnb 20979	. . . . . 6 |- ( ( ( (ordTop ` <_ ) e. Top /\ ( 0 (,] +oo ) e. _V ) /\ ( ( 0 (,] +oo ) e. (ordTop ` <_ ) /\ ( 0 (,] +oo ) C_ ( 0 (,] +oo ) /\ ( A i^i ( 0 (,] +oo ) ) C_ ( 0 (,] +oo ) ) ) -> ( ( A i^i ( 0 (,] +oo ) ) e. (ordTop ` <_ ) <-> ( A i^i ( 0 (,] +oo ) ) e. ( (ordTop ` <_ ) ↾t ( 0 (,] +oo ) ) ) )
57	49, 36, 51, 53, 55, 56	syl23anc 1333	. . . . 5 |- ( A e. J -> ( ( A i^i ( 0 (,] +oo ) ) e. (ordTop ` <_ ) <-> ( A i^i ( 0 (,] +oo ) ) e. ( (ordTop ` <_ ) ↾t ( 0 (,] +oo ) ) ) )
58	48, 57	mpbird 247	. . . 4 |- ( A e. J -> ( A i^i ( 0 (,] +oo ) ) e. (ordTop ` <_ ) )
59	58	adantr 481	. . 3 |- ( ( A e. J /\ +oo e. A ) -> ( A i^i ( 0 (,] +oo ) ) e. (ordTop ` <_ ) )
60		simpr 477	. . . 4 |- ( ( A e. J /\ +oo e. A ) -> +oo e. A )
61		0ltpnf 11956	. . . . . 6 |- 0 < +oo
62		ubioc1 12227	. . . . . 6 |- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 < +oo ) -> +oo e. ( 0 (,] +oo ) )
63	5, 7, 61, 62	mp3an 1424	. . . . 5 |- +oo e. ( 0 (,] +oo )
64	63	a1i 11	. . . 4 |- ( ( A e. J /\ +oo e. A ) -> +oo e. ( 0 (,] +oo ) )
65	60, 64	elind 3798	. . 3 |- ( ( A e. J /\ +oo e. A ) -> +oo e. ( A i^i ( 0 (,] +oo ) ) )
66		pnfnei 21024	. . 3 |- ( ( ( A i^i ( 0 (,] +oo ) ) e. (ordTop ` <_ ) /\ +oo e. ( A i^i ( 0 (,] +oo ) ) ) -> E. y e. RR ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) )
67	59, 65, 66	syl2anc 693	. 2 |- ( ( A e. J /\ +oo e. A ) -> E. y e. RR ( y (,] +oo ) C_ ( A i^i ( 0 (,] +oo ) ) )
68	28, 67	r19.29a 3078	1 |- ( ( A e. J /\ +oo e. A ) -> E. x e. RR ( x (,] +oo ) C_ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176   [,]cicc 12178   ↾_s cress 15858   ↾_t crest 16081   TopOpenctopn 16082  ordTopcordt 16159   RR*scxrs 16160   Topctop 20698  TopOnctopon 20715

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-rep 4771  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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-ps 17200  df-tsr 17201  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  lmxrge0  29998