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

Theorem ifle 10540
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 994 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  e.  RR )
21leidd 9355 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  <_  A )
3 iftrue 3584 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
43adantl 452 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  =  A )
5 id 19 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
65imp 418 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
76adantll 694 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  ps )
8 iftrue 3584 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
97, 8syl 15 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ps ,  A ,  B )  =  A )
102, 4, 93brtr4d 4069 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
11 iffalse 3585 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
1211adantl 452 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  =  B )
13 simpll3 996 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  A )
14 simpll2 995 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  e.  RR )
1514leidd 9355 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  B )
16 breq2 4043 . . . . 5  |-  ( A  =  if ( ps ,  A ,  B
)  ->  ( B  <_  A  <->  B  <_  if ( ps ,  A ,  B ) ) )
17 breq2 4043 . . . . 5  |-  ( B  =  if ( ps ,  A ,  B
)  ->  ( B  <_  B  <->  B  <_  if ( ps ,  A ,  B ) ) )
1816, 17ifboth 3609 . . . 4  |-  ( ( B  <_  A  /\  B  <_  B )  ->  B  <_  if ( ps ,  A ,  B
) )
1913, 15, 18syl2anc 642 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  if ( ps ,  A ,  B ) )
2012, 19eqbrtrd 4059 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B )
)
2110, 20pm2.61dan 766 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ifcif 3578   class class class wbr 4039   RRcr 8752    <_ cle 8884
This theorem is referenced by:  rpnnen2lem4  12512  itg2cnlem2  19133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-pre-lttri 8827
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889
  Copyright terms: Public domain W3C validator