Theorem metss2 14842

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 9773	. . . . 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 14841	. . . 4 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
11		oveq2 5933	. . . . . 6 |- ( s = ( r / R ) -> ( x ( ball ` D ) s ) = ( x ( ball ` D ) ( r / R ) ) )
12	11	sseq1d 3213	. . . . 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 2868	. . . 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 2579	. 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 14699	. . . 4 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
17	7, 16	syl 14	. . 3 |- ( ph -> C e. ( *Met ` X ) )
18		metxmet 14699	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
19	8, 18	syl 14	. . 3 |- ( ph -> D e. ( *Met ` X ) )
20	5, 6	metss 14838	. . 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 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   x. cmul 7903   <_ cle 8081   / cdiv 8718   RR+crp 9747   *Metcxmet 14170   Metcmet 14171   ballcbl 14172   MetOpencmopn 14175

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-xneg 9866  df-xadd 9867  df-seqfrec 10559  df-exp 10650  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-topgen 12964  df-psmet 14177  df-xmet 14178  df-met 14179  df-bl 14180  df-mopn 14181  df-top 14342  df-bases 14387

This theorem is referenced by: (None)