Theorem metnrmlem1a 22661

Description: Lemma for metnrm 22665. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Hypotheses

Ref	Expression
metdscn.f	|- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
metdscn.j	|- J = ( MetOpen ` D )
metnrmlem.1	|- ( ph -> D e. ( *Met ` X ) )
metnrmlem.2	|- ( ph -> S e. ( Clsd ` J ) )
metnrmlem.3	|- ( ph -> T e. ( Clsd ` J ) )
metnrmlem.4	|- ( ph -> ( S i^i T ) = (/) )

Assertion

Ref	Expression
metnrmlem1a	|- ( ( ph /\ A e. T ) -> ( 0 < ( F ` A ) /\ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR+ ) )

Distinct variable groups:   x, y, A   x, D, y   y, J   x, T, y   x, S, y   x, X, y

Allowed substitution hints:   ph( x, y)   F( x, y)   J( x)

Proof of Theorem metnrmlem1a

Step	Hyp	Ref	Expression
1		metnrmlem.4	. . . . . 6 |- ( ph -> ( S i^i T ) = (/) )
2	1	adantr 481	. . . . 5 |- ( ( ph /\ A e. T ) -> ( S i^i T ) = (/) )
3		inelcm 4032	. . . . . . . 8 |- ( ( A e. S /\ A e. T ) -> ( S i^i T ) =/= (/) )
4	3	expcom 451	. . . . . . 7 |- ( A e. T -> ( A e. S -> ( S i^i T ) =/= (/) ) )
5	4	adantl 482	. . . . . 6 |- ( ( ph /\ A e. T ) -> ( A e. S -> ( S i^i T ) =/= (/) ) )
6	5	necon2bd 2810	. . . . 5 |- ( ( ph /\ A e. T ) -> ( ( S i^i T ) = (/) -> -. A e. S ) )
7	2, 6	mpd 15	. . . 4 |- ( ( ph /\ A e. T ) -> -. A e. S )
8		eqcom 2629	. . . . . 6 |- ( 0 = ( F ` A ) <-> ( F ` A ) = 0 )
9		metnrmlem.1	. . . . . . . 8 |- ( ph -> D e. ( *Met ` X ) )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. T ) -> D e. ( *Met ` X ) )
11		metnrmlem.2	. . . . . . . . . 10 |- ( ph -> S e. ( Clsd ` J ) )
12	11	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A e. T ) -> S e. ( Clsd ` J ) )
13		eqid 2622	. . . . . . . . . 10 |- U. J = U. J
14	13	cldss 20833	. . . . . . . . 9 |- ( S e. ( Clsd ` J ) -> S C_ U. J )
15	12, 14	syl 17	. . . . . . . 8 |- ( ( ph /\ A e. T ) -> S C_ U. J )
16		metdscn.j	. . . . . . . . . 10 |- J = ( MetOpen ` D )
17	16	mopnuni 22246	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> X = U. J )
18	10, 17	syl 17	. . . . . . . 8 |- ( ( ph /\ A e. T ) -> X = U. J )
19	15, 18	sseqtr4d 3642	. . . . . . 7 |- ( ( ph /\ A e. T ) -> S C_ X )
20		metnrmlem.3	. . . . . . . . . . 11 |- ( ph -> T e. ( Clsd ` J ) )
21	20	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A e. T ) -> T e. ( Clsd ` J ) )
22	13	cldss 20833	. . . . . . . . . 10 |- ( T e. ( Clsd ` J ) -> T C_ U. J )
23	21, 22	syl 17	. . . . . . . . 9 |- ( ( ph /\ A e. T ) -> T C_ U. J )
24	23, 18	sseqtr4d 3642	. . . . . . . 8 |- ( ( ph /\ A e. T ) -> T C_ X )
25		simpr 477	. . . . . . . 8 |- ( ( ph /\ A e. T ) -> A e. T )
26	24, 25	sseldd 3604	. . . . . . 7 |- ( ( ph /\ A e. T ) -> A e. X )
27		metdscn.f	. . . . . . . 8 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
28	27, 16	metdseq0 22657	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( ( F ` A ) = 0 <-> A e. ( ( cls ` J ) ` S ) ) )
29	10, 19, 26, 28	syl3anc 1326	. . . . . 6 |- ( ( ph /\ A e. T ) -> ( ( F ` A ) = 0 <-> A e. ( ( cls ` J ) ` S ) ) )
30	8, 29	syl5bb 272	. . . . 5 |- ( ( ph /\ A e. T ) -> ( 0 = ( F ` A ) <-> A e. ( ( cls ` J ) ` S ) ) )
31		cldcls 20846	. . . . . . 7 |- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = S )
32	12, 31	syl 17	. . . . . 6 |- ( ( ph /\ A e. T ) -> ( ( cls ` J ) ` S ) = S )
33	32	eleq2d 2687	. . . . 5 |- ( ( ph /\ A e. T ) -> ( A e. ( ( cls ` J ) ` S ) <-> A e. S ) )
34	30, 33	bitrd 268	. . . 4 |- ( ( ph /\ A e. T ) -> ( 0 = ( F ` A ) <-> A e. S ) )
35	7, 34	mtbird 315	. . 3 |- ( ( ph /\ A e. T ) -> -. 0 = ( F ` A ) )
36	27	metdsf 22651	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
37	10, 19, 36	syl2anc 693	. . . . . . 7 |- ( ( ph /\ A e. T ) -> F : X --> ( 0 [,] +oo ) )
38	37, 26	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ A e. T ) -> ( F ` A ) e. ( 0 [,] +oo ) )
39		elxrge0 12281	. . . . . . 7 |- ( ( F ` A ) e. ( 0 [,] +oo ) <-> ( ( F ` A ) e. RR* /\ 0 <_ ( F ` A ) ) )
40	39	simprbi 480	. . . . . 6 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` A ) )
41	38, 40	syl 17	. . . . 5 |- ( ( ph /\ A e. T ) -> 0 <_ ( F ` A ) )
42		0xr 10086	. . . . . 6 |- 0 e. RR*
43	39	simplbi 476	. . . . . . 7 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> ( F ` A ) e. RR* )
44	38, 43	syl 17	. . . . . 6 |- ( ( ph /\ A e. T ) -> ( F ` A ) e. RR* )
45		xrleloe 11977	. . . . . 6 |- ( ( 0 e. RR* /\ ( F ` A ) e. RR* ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) )
46	42, 44, 45	sylancr 695	. . . . 5 |- ( ( ph /\ A e. T ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) )
47	41, 46	mpbid 222	. . . 4 |- ( ( ph /\ A e. T ) -> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) )
48	47	ord 392	. . 3 |- ( ( ph /\ A e. T ) -> ( -. 0 < ( F ` A ) -> 0 = ( F ` A ) ) )
49	35, 48	mt3d 140	. 2 |- ( ( ph /\ A e. T ) -> 0 < ( F ` A ) )
50		1re 10039	. . . . . 6 |- 1 e. RR
51	50	rexri 10097	. . . . 5 |- 1 e. RR*
52		ifcl 4130	. . . . 5 |- ( ( 1 e. RR* /\ ( F ` A ) e. RR* ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR* )
53	51, 44, 52	sylancr 695	. . . 4 |- ( ( ph /\ A e. T ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR* )
54		1red 10055	. . . 4 |- ( ( ph /\ A e. T ) -> 1 e. RR )
55		0lt1 10550	. . . . . 6 |- 0 < 1
56		breq2 4657	. . . . . . 7 |- ( 1 = if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) -> ( 0 < 1 <-> 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) ) )
57		breq2 4657	. . . . . . 7 |- ( ( F ` A ) = if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) -> ( 0 < ( F ` A ) <-> 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) ) )
58	56, 57	ifboth 4124	. . . . . 6 |- ( ( 0 < 1 /\ 0 < ( F ` A ) ) -> 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) )
59	55, 49, 58	sylancr 695	. . . . 5 |- ( ( ph /\ A e. T ) -> 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) )
60		xrltle 11982	. . . . . 6 |- ( ( 0 e. RR* /\ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR* ) -> ( 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) -> 0 <_ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) ) )
61	42, 53, 60	sylancr 695	. . . . 5 |- ( ( ph /\ A e. T ) -> ( 0 < if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) -> 0 <_ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) ) )
62	59, 61	mpd 15	. . . 4 |- ( ( ph /\ A e. T ) -> 0 <_ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) )
63		xrmin1 12008	. . . . 5 |- ( ( 1 e. RR* /\ ( F ` A ) e. RR* ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) <_ 1 )
64	51, 44, 63	sylancr 695	. . . 4 |- ( ( ph /\ A e. T ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) <_ 1 )
65		xrrege0 12005	. . . 4 |- ( ( ( if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR* /\ 1 e. RR ) /\ ( 0 <_ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) /\ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) <_ 1 ) ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR )
66	53, 54, 62, 64, 65	syl22anc 1327	. . 3 |- ( ( ph /\ A e. T ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR )
67	66, 59	elrpd 11869	. 2 |- ( ( ph /\ A e. T ) -> if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR+ )
68	49, 67	jca 554	1 |- ( ( ph /\ A e. T ) -> ( 0 < ( F ` A ) /\ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR+ ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   [,]cicc 12178   *Metcxmt 19731   MetOpencmopn 19736   Clsdccld 20820   clsccl 20822

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  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-iin 4523  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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  metnrmlem2  22663  metnrmlem3  22664