Theorem metss2 15599

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 10080	. . . . 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 15598	. . . 4 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
11		oveq2 6093	. . . . . 6 |- ( s = ( r / R ) -> ( x ( ball ` D ) s ) = ( x ( ball ` D ) ( r / R ) ) )
12	11	sseq1d 3277	. . . . 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 2929	. . . 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 415	. . 3 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) )
15	14	ralrimivva 2632	. 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 15456	. . . 4 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
17	7, 16	syl 14	. . 3 |- ( ph -> C e. ( *Met ` X ) )
18		metxmet 15456	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
19	8, 18	syl 14	. . 3 |- ( ph -> D e. ( *Met ` X ) )
20	5, 6	metss 15595	. . 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 415	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   x. cmul 8184   <_ cle 8361   / cdiv 9002   RR+crp 10054   *Metcxmet 14873   Metcmet 14874   ballcbl 14875   MetOpencmopn 14878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-bases 15144

This theorem is used by: (None)