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

Theorem leltadd 9929
Description: Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
Assertion
Ref Expression
leltadd  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( A  +  B )  <  ( C  +  D )
) )

Proof of Theorem leltadd
StepHypRef Expression
1 ltleadd 9928 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( B  < 
D  /\  A  <_  C )  ->  ( B  +  A )  <  ( D  +  C )
) )
21ancomsd 454 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
32ancom2s 800 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
43ancom1s 803 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
5 recn 9478 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
6 recn 9478 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
7 addcom 9661 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
85, 6, 7syl2an 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
9 recn 9478 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
10 recn 9478 . . . 4  |-  ( D  e.  RR  ->  D  e.  CC )
11 addcom 9661 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
129, 10, 11syl2an 477 . . 3  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  =  ( D  +  C ) )
138, 12breqan12d 4410 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  B )  <  ( C  +  D )  <->  ( B  +  A )  <  ( D  +  C ) ) )
144, 13sylibrd 234 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( A  +  B )  <  ( C  +  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4395  (class class class)co 6195   CCcc 9386   RRcr 9387    + caddc 9391    < clt 9524    <_ cle 9525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530
This theorem is referenced by:  lt2add  9930  addgegt0  9932  leltaddd  10066  fldiv  11811
  Copyright terms: Public domain W3C validator