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

Theorem ifle 10715
Description: An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)
Assertion
Ref Expression
ifle  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps )
)  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )

Proof of Theorem ifle
StepHypRef Expression
1 simpll1 996 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  e.  RR )
21leidd 9525 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  <_  A )
3 iftrue 3688 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
43adantl 453 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  =  A )
5 id 20 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
65imp 419 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
76adantll 695 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  ps )
8 iftrue 3688 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
97, 8syl 16 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ps ,  A ,  B )  =  A )
102, 4, 93brtr4d 4183 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
11 iffalse 3689 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
1211adantl 453 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  =  B )
13 simpll3 998 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  A )
14 simpll2 997 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  e.  RR )
1514leidd 9525 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  B )
16 breq2 4157 . . . . 5  |-  ( A  =  if ( ps ,  A ,  B
)  ->  ( B  <_  A  <->  B  <_  if ( ps ,  A ,  B ) ) )
17 breq2 4157 . . . . 5  |-  ( B  =  if ( ps ,  A ,  B
)  ->  ( B  <_  B  <->  B  <_  if ( ps ,  A ,  B ) ) )
1816, 17ifboth 3713 . . . 4  |-  ( ( B  <_  A  /\  B  <_  B )  ->  B  <_  if ( ps ,  A ,  B
) )
1913, 15, 18syl2anc 643 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  if ( ps ,  A ,  B ) )
2012, 19eqbrtrd 4173 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B )
)
2110, 20pm2.61dan 767 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps )
)  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ifcif 3682   class class class wbr 4153   RRcr 8922    <_ cle 9054
This theorem is referenced by:  rpnnen2lem4  12744  itg2cnlem2  19521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-pre-lttri 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059
  Copyright terms: Public domain W3C validator