Theorem metss2 15489

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 110	. . . . 5 |- ( ( x e. X /\ r e. RR+ ) -> r e. RR+ )
2		metss2.3	. . . . 5 |- ( ph -> R e. RR+ )
3		rpdivcl 10030	. . . . 5 |- ( ( r e. RR+ /\ R e. RR+ ) -> ( r / R ) e. RR+ )
4	1, 2, 3	syl2anr 290	. . . 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 15488	. . . 4 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
11		oveq2 6066	. . . . . 6 |- ( s = ( r / R ) -> ( x ( ball ` D ) s ) = ( x ( ball ` D ) ( r / R ) ) )
12	11	sseq1d 3271	. . . . 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 2923	. . . 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 411	. . 3 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
15	14	ralrimivva 2626	. 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 15346	. . . 4 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
17	7, 16	syl 14	. . 3 |- ( ph -> C e. ( *Met ` X ) )
18		metxmet 15346	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
19	8, 18	syl 14	. . 3 |- ( ph -> D e. ( *Met ` X ) )
20	5, 6	metss 15485	. . 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 411	. 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 167	1 |- ( ph -> J C_ K )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3214   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   x. cmul 8148   <_ cle 8325   / cdiv 8963   RR+crp 10004   *Metcxmet 14810   Metcmet 14811   ballcbl 14812   MetOpencmopn 14815

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-bases 15034

This theorem is referenced by: (None)