Theorem metss2lem 15211

Description: Lemma for metss2 15212. (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 535	. . . . . 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 15067	. . . . . 6 |- ( ( D e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR )
6	2, 3, 4, 5	syl3anc 1271	. . . . 5 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x D y ) e. RR )
7		simplrr 536	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> S e. RR+ )
8	7	rpred 9921	. . . . 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 9970	. . . 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 15067	. . . . . . 7 |- ( ( C e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR )
18	16, 3, 4, 17	syl3anc 1271	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) e. RR )
19	10	rpred 9921	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR )
20	19, 6	remulcld 8200	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( R x. ( x D y ) ) e. RR )
21		lelttr 8258	. . . . . 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 1271	. . . . 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 3306	. 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 15069	. . . . 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 529	. . 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 9904	. . . . 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 9922	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR* )
34		blval 15103	. . 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 1271	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
36		metxmet 15069	. . . . 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 9886	. . . 4 |- ( S e. RR+ -> S e. RR* )
40	39	ad2antll 491	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> S e. RR* )
41		blval 15103	. . 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 1271	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
43	25, 35, 42	3sstr4d 3270	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 1395   e. wcel 2200   {crab 2512   C_ wss 3198   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   RRcr 8021   x. cmul 8027   RR*cxr 8203   < clt 8204   <_ cle 8205   / cdiv 8842   RR+crp 9878   *Metcxmet 14540   Metcmet 14541   ballcbl 14542   MetOpencmopn 14545

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-rp 9879  df-xadd 9998  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550

This theorem is referenced by:  metss2  15212