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Theorem leltadd 8202
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 8201 . . . . 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 555 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
43ancom1s 558 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
5 recn 7746 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
6 recn 7746 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
7 addcom 7892 . . . 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 7746 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
10 recn 7746 . . . 4  |-  ( D  e.  RR  ->  D  e.  CC )
11 addcom 7892 . . . 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 3940 . 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 1331    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612    + caddc 7616    < clt 7793    <_ cle 7794
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-pre-ltwlin 7726  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-iota 5083  df-fv 5126  df-ov 5770  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799
This theorem is referenced by:  addgegt0  8204  leltaddd  8321
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