Theorem metss2lem 15362

Description: Lemma for metss2 15363. (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 15218	. . . . . 6 |- ( ( D e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR )
6	2, 3, 4, 5	syl3anc 1274	. . . . 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 10029	. . . . 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 10078	. . . 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 15218	. . . . . . 7 |- ( ( C e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x C y ) e. RR )
18	16, 3, 4, 17	syl3anc 1274	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( x C y ) e. RR )
19	10	rpred 10029	. . . . . . 7 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> R e. RR )
20	19, 6	remulcld 8304	. . . . . 6 |- ( ( ( ph /\ ( x e. X /\ S e. RR+ ) ) /\ y e. X ) -> ( R x. ( x D y ) ) e. RR )
21		lelttr 8362	. . . . . 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 1274	. . . . 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 3319	. 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 15220	. . . . 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 10012	. . . . 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 10030	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( S / R ) e. RR* )
34		blval 15254	. . 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 1274	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` D ) ( S / R ) ) = { y e. X | ( x D y ) < ( S / R ) } )
36		metxmet 15220	. . . . 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 9994	. . . 4 |- ( S e. RR+ -> S e. RR* )
40	39	ad2antll 491	. . 3 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> S e. RR* )
41		blval 15254	. . 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 1274	. 2 |- ( ( ph /\ ( x e. X /\ S e. RR+ ) ) -> ( x ( ball ` C ) S ) = { y e. X | ( x C y ) < S } )
43	25, 35, 42	3sstr4d 3283	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 1398   e. wcel 2203   {crab 2524   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   RRcr 8126   x. cmul 8132   RR*cxr 8307   < clt 8308   <_ cle 8309   / cdiv 8946   RR+crp 9986   *Metcxmet 14684   Metcmet 14685   ballcbl 14686   MetOpencmopn 14689

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-rp 9987  df-xadd 10106  df-psmet 14691  df-xmet 14692  df-met 14693  df-bl 14694

This theorem is referenced by:  metss2  15363