ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elcncf2 Unicode version

Theorem elcncf2 14521
Description: Version of elcncf 14520 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
Assertion
Ref Expression
elcncf2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
Distinct variable groups:    x, w, y, z, A    w, F, x, y, z    w, B, x, y, z

Proof of Theorem elcncf2
StepHypRef Expression
1 elcncf 14520 . 2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) ) )
2 simplll 533 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  A  C_  CC )
3 simprl 529 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  x  e.  A )
42, 3sseldd 3171 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  x  e.  CC )
5 simprr 531 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  w  e.  A )
62, 5sseldd 3171 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  w  e.  CC )
74, 6abssubd 11234 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( abs `  ( x  -  w
) )  =  ( abs `  ( w  -  x ) ) )
87breq1d 4028 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( ( abs `  ( x  -  w ) )  < 
z  <->  ( abs `  (
w  -  x ) )  <  z ) )
9 simpllr 534 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  B  C_  CC )
10 simplr 528 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  F : A
--> B )
1110, 3ffvelcdmd 5673 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( F `  x )  e.  B
)
129, 11sseldd 3171 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( F `  x )  e.  CC )
1310, 5ffvelcdmd 5673 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( F `  w )  e.  B
)
149, 13sseldd 3171 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( F `  w )  e.  CC )
1512, 14abssubd 11234 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  =  ( abs `  ( ( F `  w )  -  ( F `  x ) ) ) )
1615breq1d 4028 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( ( abs `  ( ( F `
 x )  -  ( F `  w ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  x ) ) )  <  y ) )
178, 16imbi12d 234 . . . . . . . 8  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  (
x  e.  A  /\  w  e.  A )
)  ->  ( (
( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <-> 
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
1817anassrs 400 . . . . . . 7  |-  ( ( ( ( ( A 
C_  CC  /\  B  C_  CC )  /\  F : A
--> B )  /\  x  e.  A )  /\  w  e.  A )  ->  (
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <-> 
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
1918ralbidva 2486 . . . . . 6  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  x  e.  A )  ->  ( A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <->  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
2019rexbidv 2491 . . . . 5  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  x  e.  A )  ->  ( E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  < 
z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
)  <->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
2120ralbidv 2490 . . . 4  |-  ( ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  /\  x  e.  A )  ->  ( A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <->  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
2221ralbidva 2486 . . 3  |-  ( ( ( A  C_  CC  /\  B  C_  CC )  /\  F : A --> B )  ->  ( A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
2322pm5.32da 452 . 2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  (
( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )  <->  ( F : A
--> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
241, 23bitrd 188 1  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160   A.wral 2468   E.wrex 2469    C_ wss 3144   class class class wbr 4018   -->wf 5231   ` cfv 5235  (class class class)co 5896   CCcc 7839    < clt 8022    - cmin 8158   RR+crp 9683   abscabs 11038   -cn->ccncf 14517
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-map 6676  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-2 9008  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-cncf 14518
This theorem is referenced by:  cncfi  14525  cncfcdm  14529  abscncf  14532  recncf  14533  imcncf  14534  cjcncf  14535  mulc1cncf  14536  cncfco  14538  cdivcncfap  14547  mulcncf  14551
  Copyright terms: Public domain W3C validator