Theorem metss2lem 22316

Description: Lemma for metss2 22317. (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
metequiv.3	|- J = ( MetOpen ` C )
metequiv.4	|- K = ( MetOpen ` D )
metss2.1	|- ( ph -> C e. ( Met ` X ) )
metss2.2	|- ( ph -> D e. ( Met ` X ) )
metss2.3	|- ( ph -> R e. RR+ )
metss2.4	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )

Assertion

Ref	Expression
metss2lem	|- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) C_ ( x ( ball ` C ) S ) )

Distinct variable groups:   x, y, C   x, J, y   x, K, y   y, R   y, S   x, D, y   ph, x, y   x, X, y

Allowed substitution hints:   R( x)   S( x)

Proof of Theorem metss2lem

Step	Hyp	Ref	Expression
1		metss2.2	. . . . . . 7 |- ( ph -> D e. ( Met ` X ) )
2	1	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> D e. ( Met ` X ) )
3		simplrl 800	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> x e. X )
4		simpr 477	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> y e. X )
5		metcl 22137	. . . . . 6 |- ( ( D e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR )
6	2, 3, 4, 5	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x D y ) e. RR )
7		simplrr 801	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> S e. RR+ )
8	7	rpred 11872	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> S e. RR )
9		metss2.3	. . . . . 6 |- ( ph -> R e. RR+ )
10	9	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR+ )
11	6, 8, 10	ltmuldiv2d 11920	. . . 4 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( R x. ( x D y ) ) < S <-> ( x D y ) < ( S / R ) ) )
12		metss2.4	. . . . . . 7 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
13	12	anassrs 680	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ y e. X ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
14	13	adantlrr 757	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
15		metss2.1	. . . . . . . 8 |- ( ph -> C e. ( Met ` X ) )
16	15	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> C e. ( Met ` X ) )
17		metcl 22137	. . . . . . 7 |- ( ( C e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR )
18	16, 3, 4, 17	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) e. RR )
19	10	rpred 11872	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR )
20	19, 6	remulcld 10070	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( R x. ( x D y ) ) e. RR )
21		lelttr 10128	. . . . . 6 |- ( ( ( x C y ) e. RR /\ ( R x. ( x D y ) ) e. RR /\ S e. RR ) -> ( ( ( x C y ) <_ ( R x. ( x D y ) ) /\ ( R x. ( x D y ) ) < S ) -> ( x C y ) < S ) )
22	18, 20, 8, 21	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( ( x C y ) <_ ( R x. ( x D y ) ) /\ ( R x. ( x D y ) ) < S ) -> ( x C y ) < S ) )
23	14, 22	mpand 711	. . . 4 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( R x. ( x D y ) ) < S -> ( x C y ) < S ) )
24	11, 23	sylbird 250	. . 3 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( x D y ) < ( S / R ) -> ( x C y ) < S ) )
25	24	ss2rabdv 3683	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> { y e. X | ( x D y ) < ( S / R ) } C_ { y e. X | ( x C y ) < S } )
26		metxmet 22139	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
27	1, 26	syl 17	. . . 4 |- ( ph -> D e. ( *Met ` X ) )
28	27	adantr 481	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> D e. ( *Met ` X ) )
29		simprl 794	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> x e. X )
30		simpr 477	. . . . 5 |- ( ( x e. X /\ S e. RR+ ) -> S e. RR+ )
31		rpdivcl 11856	. . . . 5 |- ( ( S e. RR+ /\ R e. RR+ ) -> ( S / R ) e. RR+ )
32	30, 9, 31	syl2anr 495	. . . 4 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR+ )
33	32	rpxrd 11873	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR* )
34		blval 22191	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ ( S / R ) e. RR* ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
35	28, 29, 33, 34	syl3anc 1326	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
36		metxmet 22139	. . . . 5 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
37	15, 36	syl 17	. . . 4 |- ( ph -> C e. ( *Met ` X ) )
38	37	adantr 481	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> C e. ( *Met ` X ) )
39		rpxr 11840	. . . 4 |- ( S e. RR+ -> S e. RR* )
40	39	ad2antll 765	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> S e. RR* )
41		blval 22191	. . 3 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ S e. RR* ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
42	38, 29, 40, 41	syl3anc 1326	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
43	25, 35, 42	3sstr4d 3648	1 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) C_ ( x ( ball ` C ) S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   RR+crp 11832   *Metcxmt 19731   Metcme 19732   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

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-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-rp 11833  df-xadd 11947  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741

This theorem is referenced by:  metss2  22317  equivcfil  23097  equivcau  23098  equivtotbnd  33577