Theorem metdscnlem 22658

Description: Lemma for metdscn 22659. (Contributed 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 )
metdscn.c	|- C = ( dist ` RR*s )
metdscn.k	|- K = ( MetOpen ` C )
metdscnlem.1	|- ( ph -> D e. ( *Met ` X ) )
metdscnlem.2	|- ( ph -> S C_ X )
metdscnlem.3	|- ( ph -> A e. X )
metdscnlem.4	|- ( ph -> B e. X )
metdscnlem.5	|- ( ph -> R e. RR+ )
metdscnlem.6	|- ( ph -> ( A D B ) < R )

Assertion

Ref	Expression
metdscnlem	|- ( ph -> ( ( F ` A ) +e -e ( F ` B ) ) < R )

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

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

Proof of Theorem metdscnlem

Step	Hyp	Ref	Expression
1		metdscnlem.1	. . . . . 6 |- ( ph -> D e. ( *Met ` X ) )
2		metdscnlem.2	. . . . . 6 |- ( ph -> S C_ X )
3		metdscn.f	. . . . . . 7 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
4	3	metdsf 22651	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
5	1, 2, 4	syl2anc 693	. . . . 5 |- ( ph -> F : X --> ( 0 [,] +oo ) )
6		metdscnlem.3	. . . . 5 |- ( ph -> A e. X )
7	5, 6	ffvelrnd 6360	. . . 4 |- ( ph -> ( F ` A ) e. ( 0 [,] +oo ) )
8		elxrge0 12281	. . . . 5 |- ( ( F ` A ) e. ( 0 [,] +oo ) <-> ( ( F ` A ) e. RR* /\ 0 <_ ( F ` A ) ) )
9	8	simplbi 476	. . . 4 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> ( F ` A ) e. RR* )
10	7, 9	syl 17	. . 3 |- ( ph -> ( F ` A ) e. RR* )
11		metdscnlem.4	. . . . . 6 |- ( ph -> B e. X )
12	5, 11	ffvelrnd 6360	. . . . 5 |- ( ph -> ( F ` B ) e. ( 0 [,] +oo ) )
13		elxrge0 12281	. . . . . 6 |- ( ( F ` B ) e. ( 0 [,] +oo ) <-> ( ( F ` B ) e. RR* /\ 0 <_ ( F ` B ) ) )
14	13	simplbi 476	. . . . 5 |- ( ( F ` B ) e. ( 0 [,] +oo ) -> ( F ` B ) e. RR* )
15	12, 14	syl 17	. . . 4 |- ( ph -> ( F ` B ) e. RR* )
16	15	xnegcld 12130	. . 3 |- ( ph -> -e ( F ` B ) e. RR* )
17	10, 16	xaddcld 12131	. 2 |- ( ph -> ( ( F ` A ) +e -e ( F ` B ) ) e. RR* )
18		xmetcl 22136	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
19	1, 6, 11, 18	syl3anc 1326	. 2 |- ( ph -> ( A D B ) e. RR* )
20		metdscnlem.5	. . 3 |- ( ph -> R e. RR+ )
21	20	rpxrd 11873	. 2 |- ( ph -> R e. RR* )
22	3	metdstri 22654	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( F ` A ) <_ ( ( A D B ) +e ( F ` B ) ) )
23	1, 2, 6, 11, 22	syl22anc 1327	. . 3 |- ( ph -> ( F ` A ) <_ ( ( A D B ) +e ( F ` B ) ) )
24	8	simprbi 480	. . . . 5 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` A ) )
25	7, 24	syl 17	. . . 4 |- ( ph -> 0 <_ ( F ` A ) )
26	13	simprbi 480	. . . . . 6 |- ( ( F ` B ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` B ) )
27	12, 26	syl 17	. . . . 5 |- ( ph -> 0 <_ ( F ` B ) )
28		ge0nemnf 12004	. . . . 5 |- ( ( ( F ` B ) e. RR* /\ 0 <_ ( F ` B ) ) -> ( F ` B ) =/= -oo )
29	15, 27, 28	syl2anc 693	. . . 4 |- ( ph -> ( F ` B ) =/= -oo )
30		xmetge0 22149	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
31	1, 6, 11, 30	syl3anc 1326	. . . 4 |- ( ph -> 0 <_ ( A D B ) )
32		xlesubadd 12093	. . . 4 |- ( ( ( ( F ` A ) e. RR* /\ ( F ` B ) e. RR* /\ ( A D B ) e. RR* ) /\ ( 0 <_ ( F ` A ) /\ ( F ` B ) =/= -oo /\ 0 <_ ( A D B ) ) ) -> ( ( ( F ` A ) +e -e ( F ` B ) ) <_ ( A D B ) <-> ( F ` A ) <_ ( ( A D B ) +e ( F ` B ) ) ) )
33	10, 15, 19, 25, 29, 31, 32	syl33anc 1341	. . 3 |- ( ph -> ( ( ( F ` A ) +e -e ( F ` B ) ) <_ ( A D B ) <-> ( F ` A ) <_ ( ( A D B ) +e ( F ` B ) ) ) )
34	23, 33	mpbird 247	. 2 |- ( ph -> ( ( F ` A ) +e -e ( F ` B ) ) <_ ( A D B ) )
35		metdscnlem.6	. 2 |- ( ph -> ( A D B ) < R )
36	17, 19, 21, 34, 35	xrlelttrd 11991	1 |- ( ph -> ( ( F ` A ) +e -e ( F ` B ) ) < R )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   0cc0 9936   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   -ecxne 11943   +ecxad 11944   [,]cicc 12178   distcds 15950   RR*scxrs 16160   *Metcxmt 19731   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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  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-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-psmet 19738  df-xmet 19739  df-bl 19741

This theorem is referenced by:  metdscn  22659