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Theorem leltadd 8322
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 8321 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( B  < 
D  /\  A  <_  C )  ->  ( B  +  A )  <  ( D  +  C )
) )
21ancomsd 267 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
32ancom2s 556 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
43ancom1s 559 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
5 recn 7865 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
6 recn 7865 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
7 addcom 8012 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
85, 6, 7syl2an 287 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
9 recn 7865 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
10 recn 7865 . . . 4  |-  ( D  e.  RR  ->  D  e.  CC )
11 addcom 8012 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
129, 10, 11syl2an 287 . . 3  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  =  ( D  +  C ) )
138, 12breqan12d 3981 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  B )  <  ( C  +  D )  <->  ( B  +  A )  <  ( D  +  C ) ) )
144, 13sylibrd 168 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 set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   CCcc 7730   RRcr 7731    + caddc 7735    < clt 7912    <_ cle 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-pre-ltwlin 7845  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-cnv 4594  df-iota 5135  df-fv 5178  df-ov 5827  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918
This theorem is referenced by:  addgegt0  8324  leltaddd  8441
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