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Theorem leltadd 8402
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 8401 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( B  < 
D  /\  A  <_  C )  ->  ( B  +  A )  <  ( D  +  C )
) )
21ancomsd 269 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
32ancom2s 566 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
43ancom1s 569 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
5 recn 7943 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
6 recn 7943 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
7 addcom 8092 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
85, 6, 7syl2an 289 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
9 recn 7943 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
10 recn 7943 . . . 4  |-  ( D  e.  RR  ->  D  e.  CC )
11 addcom 8092 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
129, 10, 11syl2an 289 . . 3  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  =  ( D  +  C ) )
138, 12breqan12d 4019 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  B )  <  ( C  +  D )  <->  ( B  +  A )  <  ( D  +  C ) ) )
144, 13sylibrd 169 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 104    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808   RRcr 7809    + caddc 7813    < clt 7990    <_ cle 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-pre-ltwlin 7923  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-iota 5178  df-fv 5224  df-ov 5877  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996
This theorem is referenced by:  addgegt0  8404  leltaddd  8521
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