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Theorem metcnp2 18088
Description: Two ways to say a mapping from metric  C to metric  D is continuous at point  P. The distance arguments are swapped compared to metcnp 18087 (and Munkres' metcn 18089) for compatibility with df-lm 16959. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypotheses
Ref Expression
metcn.2  |-  J  =  ( MetOpen `  C )
metcn.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metcnp2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  < 
z  ->  ( ( F `  w ) D ( F `  P ) )  < 
y ) ) ) )
Distinct variable groups:    y, w, z, F    w, J, y, z    w, K, y, z    w, X, y, z    w, Y, y, z    w, C, y, z    w, D, y, z    w, P, y, z

Proof of Theorem metcnp2
StepHypRef Expression
1 metcn.2 . . 3  |-  J  =  ( MetOpen `  C )
2 metcn.4 . . 3  |-  K  =  ( MetOpen `  D )
31, 2metcnp 18087 . 2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  < 
z  ->  ( ( F `  P ) D ( F `  w ) )  < 
y ) ) ) )
4 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  ->  C  e.  ( * Met `  X ) )
54ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  C  e.  ( * Met `  X
) )
6 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  ->  P  e.  X )
76ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  P  e.  X )
8 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  w  e.  X )
9 xmetsym 17912 . . . . . . . . . 10  |-  ( ( C  e.  ( * Met `  X )  /\  P  e.  X  /\  w  e.  X
)  ->  ( P C w )  =  ( w C P ) )
105, 7, 8, 9syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  ( P C w )  =  ( w C P ) )
1110breq1d 4033 . . . . . . . 8  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  (
( P C w )  <  z  <->  ( w C P )  <  z
) )
12 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  ->  D  e.  ( * Met `  Y ) )
1312ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  D  e.  ( * Met `  Y
) )
14 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  F : X --> Y )
15 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : X --> Y  /\  P  e.  X )  ->  ( F `  P
)  e.  Y )
1614, 7, 15syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  ( F `  P )  e.  Y )
17 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : X --> Y  /\  w  e.  X )  ->  ( F `  w
)  e.  Y )
1814, 8, 17syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  ( F `  w )  e.  Y )
19 xmetsym 17912 . . . . . . . . . 10  |-  ( ( D  e.  ( * Met `  Y )  /\  ( F `  P )  e.  Y  /\  ( F `  w
)  e.  Y )  ->  ( ( F `
 P ) D ( F `  w
) )  =  ( ( F `  w
) D ( F `
 P ) ) )
2013, 16, 18, 19syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  (
( F `  P
) D ( F `
 w ) )  =  ( ( F `
 w ) D ( F `  P
) ) )
2120breq1d 4033 . . . . . . . 8  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  (
( ( F `  P ) D ( F `  w ) )  <  y  <->  ( ( F `  w ) D ( F `  P ) )  < 
y ) )
2211, 21imbi12d 311 . . . . . . 7  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  /\  w  e.  X )  ->  (
( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y )  <-> 
( ( w C P )  <  z  ->  ( ( F `  w ) D ( F `  P ) )  <  y ) ) )
2322ralbidva 2559 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  ( y  e.  RR+  /\  z  e.  RR+ )
)  ->  ( A. w  e.  X  (
( P C w )  <  z  -> 
( ( F `  P ) D ( F `  w ) )  <  y )  <->  A. w  e.  X  ( ( w C P )  <  z  ->  ( ( F `  w ) D ( F `  P ) )  <  y ) ) )
2423anassrs 629 . . . . 5  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  y  e.  RR+ )  /\  z  e.  RR+ )  -> 
( A. w  e.  X  ( ( P C w )  < 
z  ->  ( ( F `  P ) D ( F `  w ) )  < 
y )  <->  A. w  e.  X  ( (
w C P )  <  z  ->  (
( F `  w
) D ( F `
 P ) )  <  y ) ) )
2524rexbidva 2560 . . . 4  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  /\  y  e.  RR+ )  -> 
( E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y )  <->  E. z  e.  RR+  A. w  e.  X  ( (
w C P )  <  z  ->  (
( F `  w
) D ( F `
 P ) )  <  y ) ) )
2625ralbidva 2559 . . 3  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  /\  F : X --> Y )  -> 
( A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  -> 
( ( F `  P ) D ( F `  w ) )  <  y )  <->  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  < 
z  ->  ( ( F `  w ) D ( F `  P ) )  < 
y ) ) )
2726pm5.32da 622 . 2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  ->  (
( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  -> 
( ( F `  P ) D ( F `  w ) )  <  y ) )  <->  ( F : X
--> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  <  z  ->  ( ( F `  w ) D ( F `  P ) )  <  y ) ) ) )
283, 27bitrd 244 1  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  < 
z  ->  ( ( F `  w ) D ( F `  P ) )  < 
y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    < clt 8867   RR+crp 10354   * Metcxmt 16369   MetOpencmopn 16372    CnP ccnp 16955
This theorem is referenced by:  metcnpi2  18091  rlimcnp  20260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cnp 16958
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