Theorem metnrmlem1 22662

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
metnrmlem1	|- ( ( ph /\ ( A e. S /\ B e. T ) ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) <_ ( A D B ) )

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

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

Proof of Theorem metnrmlem1

Step	Hyp	Ref	Expression
1		1re 10039	. . . 4 |- 1 e. RR
2	1	rexri 10097	. . 3 |- 1 e. RR*
3		metnrmlem.1	. . . . . . 7 |- ( ph -> D e. ( *Met ` X ) )
4	3	adantr 481	. . . . . 6 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> D e. ( *Met ` X ) )
5		metnrmlem.2	. . . . . . . . 9 |- ( ph -> S e. ( Clsd ` J ) )
6	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> S e. ( Clsd ` J ) )
7		eqid 2622	. . . . . . . . 9 |- U. J = U. J
8	7	cldss 20833	. . . . . . . 8 |- ( S e. ( Clsd ` J ) -> S C_ U. J )
9	6, 8	syl 17	. . . . . . 7 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> S C_ U. J )
10		metdscn.j	. . . . . . . . 9 |- J = ( MetOpen ` D )
11	10	mopnuni 22246	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> X = U. J )
12	4, 11	syl 17	. . . . . . 7 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> X = U. J )
13	9, 12	sseqtr4d 3642	. . . . . 6 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> S C_ X )
14		metdscn.f	. . . . . . 7 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
15	14	metdsf 22651	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
16	4, 13, 15	syl2anc 693	. . . . 5 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> F : X --> ( 0 [,] +oo ) )
17		metnrmlem.3	. . . . . . . . 9 |- ( ph -> T e. ( Clsd ` J ) )
18	17	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> T e. ( Clsd ` J ) )
19	7	cldss 20833	. . . . . . . 8 |- ( T e. ( Clsd ` J ) -> T C_ U. J )
20	18, 19	syl 17	. . . . . . 7 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> T C_ U. J )
21	20, 12	sseqtr4d 3642	. . . . . 6 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> T C_ X )
22		simprr 796	. . . . . 6 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> B e. T )
23	21, 22	sseldd 3604	. . . . 5 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> B e. X )
24	16, 23	ffvelrnd 6360	. . . 4 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( F ` B ) e. ( 0 [,] +oo ) )
25		elxrge0 12281	. . . . 5 |- ( ( F ` B ) e. ( 0 [,] +oo ) <-> ( ( F ` B ) e. RR* /\ 0 <_ ( F ` B ) ) )
26	25	simplbi 476	. . . 4 |- ( ( F ` B ) e. ( 0 [,] +oo ) -> ( F ` B ) e. RR* )
27	24, 26	syl 17	. . 3 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( F ` B ) e. RR* )
28		ifcl 4130	. . 3 |- ( ( 1 e. RR* /\ ( F ` B ) e. RR* ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) e. RR* )
29	2, 27, 28	sylancr 695	. 2 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) e. RR* )
30		simprl 794	. . . 4 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> A e. S )
31	13, 30	sseldd 3604	. . 3 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> A e. X )
32		xmetcl 22136	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
33	4, 31, 23, 32	syl3anc 1326	. 2 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( A D B ) e. RR* )
34		xrmin2 12009	. . 3 |- ( ( 1 e. RR* /\ ( F ` B ) e. RR* ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) <_ ( F ` B ) )
35	2, 27, 34	sylancr 695	. 2 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) <_ ( F ` B ) )
36	14	metdstri 22654	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( B e. X /\ A e. X ) ) -> ( F ` B ) <_ ( ( B D A ) +e ( F ` A ) ) )
37	4, 13, 23, 31, 36	syl22anc 1327	. . 3 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( F ` B ) <_ ( ( B D A ) +e ( F ` A ) ) )
38		xmetsym 22152	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) = ( A D B ) )
39	4, 23, 31, 38	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( B D A ) = ( A D B ) )
40	14	metds0 22653	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) = 0 )
41	4, 13, 30, 40	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( F ` A ) = 0 )
42	39, 41	oveq12d 6668	. . . 4 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( ( B D A ) +e ( F ` A ) ) = ( ( A D B ) +e 0 ) )
43		xaddid1 12072	. . . . 5 |- ( ( A D B ) e. RR* -> ( ( A D B ) +e 0 ) = ( A D B ) )
44	33, 43	syl 17	. . . 4 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( ( A D B ) +e 0 ) = ( A D B ) )
45	42, 44	eqtrd 2656	. . 3 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( ( B D A ) +e ( F ` A ) ) = ( A D B ) )
46	37, 45	breqtrd 4679	. 2 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> ( F ` B ) <_ ( A D B ) )
47	29, 27, 33, 35, 46	xrletrd 11993	1 |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) <_ ( A D B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   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   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   +ecxad 11944   [,]cicc 12178   *Metcxmt 19731   MetOpencmopn 19736   Clsdccld 20820

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-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-er 7742  df-ec 7744  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

This theorem is referenced by:  metnrmlem3  22664