MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfle2 Structured version   Unicode version

Theorem dfle2 11354
Description: Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
Assertion
Ref Expression
dfle2  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )

Proof of Theorem dfle2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lerel 9652 . 2  |-  Rel  <_
2 ltrelxr 9649 . . . 4  |-  <  C_  ( RR*  X.  RR* )
3 f1oi 5851 . . . . 5  |-  (  _I  |`  RR* ) : RR* -1-1-onto-> RR*
4 f1of 5816 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR*
-1-1-onto-> RR* 
->  (  _I  |`  RR* ) : RR* --> RR* )
5 fssxp 5743 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR* --> RR*  ->  (  _I  |` 
RR* )  C_  ( RR*  X.  RR* ) )
63, 4, 5mp2b 10 . . . 4  |-  (  _I  |`  RR* )  C_  ( RR*  X.  RR* )
72, 6unssi 3679 . . 3  |-  (  < 
u.  (  _I  |`  RR* )
)  C_  ( RR*  X. 
RR* )
8 relxp 5110 . . 3  |-  Rel  ( RR*  X.  RR* )
9 relss 5090 . . 3  |-  ( (  <  u.  (  _I  |`  RR* ) )  C_  ( RR*  X.  RR* )  ->  ( Rel  ( RR*  X. 
RR* )  ->  Rel  (  <  u.  (  _I  |`  RR* ) ) ) )
107, 8, 9mp2 9 . 2  |-  Rel  (  <  u.  (  _I  |`  RR* )
)
11 lerelxr 9651 . . . 4  |-  <_  C_  ( RR*  X.  RR* )
1211brel 5048 . . 3  |-  ( x  <_  y  ->  (
x  e.  RR*  /\  y  e.  RR* ) )
137brel 5048 . . 3  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  ->  ( x  e. 
RR*  /\  y  e.  RR* ) )
14 xrleloe 11351 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x  =  y ) ) )
15 resieq 5284 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (  _I  |`  RR* )
y  <->  x  =  y
) )
1615orbi2d 701 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x  <  y  \/  x (  _I  |`  RR* )
y )  <->  ( x  <  y  \/  x  =  y ) ) )
1714, 16bitr4d 256 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x (  _I  |`  RR* ) y ) ) )
18 brun 4495 . . . 4  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  <-> 
( x  <  y  \/  x (  _I  |`  RR* )
y ) )
1917, 18syl6bbr 263 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y ) )
2012, 13, 19pm5.21nii 353 . 2  |-  ( x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y )
211, 10, 20eqbrriv 5098 1  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   class class class wbr 4447    _I cid 4790    X. cxp 4997    |` cres 5001   Rel wrel 5004   -->wf 5584   -1-1-onto->wf1o 5587   RR*cxr 9628    < clt 9629    <_ cle 9630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator