Theorem metss2 12667

Description: If the metric D is "strongly finer" than C (meaning that there is a positive real constant R such that C ( x , y ) <_ R x. D ( x , y )), then D generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (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
metss2	|- ( ph -> J C_ K )

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

Allowed substitution hint:   R( x)

Proof of Theorem metss2

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 |- ( ( x e. X /\ r e. RR+ ) -> r e. RR+ )
2		metss2.3	. . . . 5 |- ( ph -> R e. RR+ )
3		rpdivcl 9467	. . . . 5 |- ( ( r e. RR+ /\ R e. RR+ ) -> ( r / R ) e. RR+ )
4	1, 2, 3	syl2anr 288	. . . 4 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( r / R ) e. RR+ )
5		metequiv.3	. . . . 5 |- J = ( MetOpen ` C )
6		metequiv.4	. . . . 5 |- K = ( MetOpen ` D )
7		metss2.1	. . . . 5 |- ( ph -> C e. ( Met ` X ) )
8		metss2.2	. . . . 5 |- ( ph -> D e. ( Met ` X ) )
9		metss2.4	. . . . 5 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
10	5, 6, 7, 8, 2, 9	metss2lem 12666	. . . 4 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
11		oveq2 5782	. . . . . 6 |- ( s = ( r / R ) -> ( x ( ball ` D ) s ) = ( x ( ball ` D ) ( r / R ) ) )
12	11	sseq1d 3126	. . . . 5 |- ( s = ( r / R ) -> ( ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) <-> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) ) )
13	12	rspcev 2789	. . . 4 |- ( ( ( r / R ) e. RR+ /\ ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) ) -> E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
14	4, 10, 13	syl2anc 408	. . 3 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
15	14	ralrimivva 2514	. 2 |- ( ph -> A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
16		metxmet 12524	. . . 4 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
17	7, 16	syl 14	. . 3 |- ( ph -> C e. ( *Met ` X ) )
18		metxmet 12524	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
19	8, 18	syl 14	. . 3 |- ( ph -> D e. ( *Met ` X ) )
20	5, 6	metss 12663	. . 3 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J C_ K <-> A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) ) )
21	17, 19, 20	syl2anc 408	. 2 |- ( ph -> ( J C_ K <-> A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) ) )
22	15, 21	mpbird 166	1 |- ( ph -> J C_ K )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   x. cmul 7625   <_ 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-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  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  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  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-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  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-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  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-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-bases 12210

This theorem is referenced by: (None)