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Theorem ltleadd 7617
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 7600 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
213com23 1145 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
323expa 1139 . . . 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 7602 . . . . . 6  |-  ( ( B  e.  RR  /\  D  e.  RR  /\  C  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
653com23 1145 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  D  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
763expb 1140 . . . 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 7161 . . . 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 7161 . . . . 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 7161 . . . 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 7267 . . 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 1170 . 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 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042    + caddc 7046    < clt 7215    <_ cle 7216
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-pre-ltwlin 7151  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-iota 4897  df-fv 4940  df-ov 5546  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221
This theorem is referenced by:  leltadd  7618  addgtge0  7621  ltleaddd  7732
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