Theorem metss2lem 12666

Description: Lemma for metss2 12667. (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 479	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> D e. ( Met ` X ) )
3		simplrl 524	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> x e. X )
4		simpr 109	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> y e. X )
5		metcl 12522	. . . . . 6 |- ( ( D e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR )
6	2, 3, 4, 5	syl3anc 1216	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x D y ) e. RR )
7		simplrr 525	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> S e. RR+ )
8	7	rpred 9483	. . . . 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 479	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR+ )
11	6, 8, 10	ltmuldiv2d 9532	. . . 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 397	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ y e. X ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
14	13	adantlrr 474	. . . . 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 479	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> C e. ( Met ` X ) )
17		metcl 12522	. . . . . . 7 |- ( ( C e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR )
18	16, 3, 4, 17	syl3anc 1216	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) e. RR )
19	10	rpred 9483	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR )
20	19, 6	remulcld 7796	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( R x. ( x D y ) ) e. RR )
21		lelttr 7852	. . . . . 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 1216	. . . . 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 425	. . . 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 169	. . 3 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( x D y ) < ( S / R ) -> ( x C y ) < S ) )
25	24	ss2rabdv 3178	. 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 12524	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
27	1, 26	syl 14	. . . 4 |- ( ph -> D e. ( *Met ` X ) )
28	27	adantr 274	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> D e. ( *Met ` X ) )
29		simprl 520	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> x e. X )
30		simpr 109	. . . . 5 |- ( ( x e. X /\ S e. RR+ ) -> S e. RR+ )
31		rpdivcl 9467	. . . . 5 |- ( ( S e. RR+ /\ R e. RR+ ) -> ( S / R ) e. RR+ )
32	30, 9, 31	syl2anr 288	. . . 4 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR+ )
33	32	rpxrd 9484	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR* )
34		blval 12558	. . 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 1216	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
36		metxmet 12524	. . . . 5 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
37	15, 36	syl 14	. . . 4 |- ( ph -> C e. ( *Met ` X ) )
38	37	adantr 274	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> C e. ( *Met ` X ) )
39		rpxr 9449	. . . 4 |- ( S e. RR+ -> S e. RR* )
40	39	ad2antll 482	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> S e. RR* )
41		blval 12558	. . 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 1216	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
43	25, 35, 42	3sstr4d 3142	1 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) C_ ( x ( ball ` C ) S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {crab 2420   C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7619   x. cmul 7625   RR*cxr 7799   < clt 7800   <_ cle 7801   / cdiv 8432   RR+crp 9441   *Metcxmet 12149   Metcmet 12150   ballcbl 12151   MetOpencmopn 12154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-rp 9442  df-xadd 9560  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159

This theorem is referenced by:  metss2  12667