Theorem metss2lem 15220

Description: Lemma for metss2 15221. (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 488	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> D e. ( Met ` X ) )
3		simplrl 537	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> x e. X )
4		simpr 110	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> y e. X )
5		metcl 15076	. . . . . 6 |- ( ( D e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR )
6	2, 3, 4, 5	syl3anc 1273	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x D y ) e. RR )
7		simplrr 538	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> S e. RR+ )
8	7	rpred 9930	. . . . 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 488	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR+ )
11	6, 8, 10	ltmuldiv2d 9979	. . . 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 400	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ y e. X ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
14	13	adantlrr 483	. . . . 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 488	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> C e. ( Met ` X ) )
17		metcl 15076	. . . . . . 7 |- ( ( C e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR )
18	16, 3, 4, 17	syl3anc 1273	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) e. RR )
19	10	rpred 9930	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR )
20	19, 6	remulcld 8209	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( R x. ( x D y ) ) e. RR )
21		lelttr 8267	. . . . . 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 1273	. . . . 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 429	. . . 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 170	. . 3 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( ( x D y ) < ( S / R ) -> ( x C y ) < S ) )
25	24	ss2rabdv 3308	. 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 15078	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
27	1, 26	syl 14	. . . 4 |- ( ph -> D e. ( *Met ` X ) )
28	27	adantr 276	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> D e. ( *Met ` X ) )
29		simprl 531	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> x e. X )
30		simpr 110	. . . . 5 |- ( ( x e. X /\ S e. RR+ ) -> S e. RR+ )
31		rpdivcl 9913	. . . . 5 |- ( ( S e. RR+ /\ R e. RR+ ) -> ( S / R ) e. RR+ )
32	30, 9, 31	syl2anr 290	. . . 4 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR+ )
33	32	rpxrd 9931	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR* )
34		blval 15112	. . 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 1273	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
36		metxmet 15078	. . . . 5 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
37	15, 36	syl 14	. . . 4 |- ( ph -> C e. ( *Met ` X ) )
38	37	adantr 276	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> C e. ( *Met ` X ) )
39		rpxr 9895	. . . 4 |- ( S e. RR+ -> S e. RR* )
40	39	ad2antll 491	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> S e. RR* )
41		blval 15112	. . 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 1273	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
43	25, 35, 42	3sstr4d 3272	1 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) C_ ( x ( ball ` C ) S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   x. cmul 8036   RR*cxr 8212   < clt 8213   <_ cle 8214   / cdiv 8851   RR+crp 9887   *Metcxmet 14549   Metcmet 14550   ballcbl 14551   MetOpencmopn 14554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-rp 9888  df-xadd 10007  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559

This theorem is referenced by:  metss2  15221