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Theorem le2add 7920
Description: Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
le2add  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <_  D
)  ->  ( A  +  B )  <_  ( C  +  D )
) )

Proof of Theorem le2add
StepHypRef Expression
1 simpll 496 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
2 simprl 498 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
3 simplr 497 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
4 leadd1 7906 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <_  C  <->  ( A  +  B )  <_  ( C  +  B )
) )
51, 2, 3, 4syl3anc 1174 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <_  C  <->  ( A  +  B )  <_  ( C  +  B ) ) )
6 simprr 499 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
7 leadd2 7907 . . . 4  |-  ( ( B  e.  RR  /\  D  e.  RR  /\  C  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
83, 6, 2, 7syl3anc 1174 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D ) ) )
95, 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 )
) ) )
101, 3readdcld 7515 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  B
)  e.  RR )
112, 3readdcld 7515 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  B
)  e.  RR )
122, 6readdcld 7515 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  D
)  e.  RR )
13 letr 7566 . . 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
) ) )
1410, 11, 12, 13syl3anc 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 )
) )
159, 14sylbid 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    <_ 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:  addge0  7927  le2addi  7987  le2addd  8038
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