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Theorem ltleadd 7922
Description: Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
Assertion
Ref Expression
ltleadd  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  B  <_  D )  ->  ( A  +  B )  <  ( C  +  D )
) )

Proof of Theorem ltleadd
StepHypRef Expression
1 ltadd1 7905 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
213com23 1149 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
323expa 1143 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  < 
C  <->  ( A  +  B )  <  ( C  +  B )
) )
43adantrr 463 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  C  <->  ( A  +  B )  <  ( C  +  B ) ) )
5 leadd2 7907 . . . . . 6  |-  ( ( B  e.  RR  /\  D  e.  RR  /\  C  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
653com23 1149 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  D  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
763expb 1144 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
87adantll 460 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D ) ) )
94, 8anbi12d 457 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  B  <_  D )  <->  ( ( A  +  B )  < 
( C  +  B
)  /\  ( C  +  B )  <_  ( C  +  D )
) ) )
10 readdcl 7466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
1110adantr 270 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  B
)  e.  RR )
12 readdcl 7466 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1312ancoms 264 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B
)  e.  RR )
1413ad2ant2lr 494 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  B
)  e.  RR )
15 readdcl 7466 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  e.  RR )
1615adantl 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  D
)  e.  RR )
17 ltletr 7572 . . 3  |-  ( ( ( A  +  B
)  e.  RR  /\  ( C  +  B
)  e.  RR  /\  ( C  +  D
)  e.  RR )  ->  ( ( ( A  +  B )  <  ( C  +  B )  /\  ( C  +  B )  <_  ( C  +  D
) )  ->  ( A  +  B )  <  ( C  +  D
) ) )
1811, 14, 16, 17syl3anc 1174 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( ( A  +  B )  < 
( C  +  B
)  /\  ( C  +  B )  <_  ( C  +  D )
)  ->  ( A  +  B )  <  ( C  +  D )
) )
199, 18sylbid 148 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 102    <-> wb 103    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7347    + caddc 7351    < clt 7520    <_ cle 7521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-pre-ltwlin 7456  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-iota 4980  df-fv 5023  df-ov 5655  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526
This theorem is referenced by:  leltadd  7923  addgtge0  7926  ltleaddd  8040
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