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Theorem ltleadd 8433
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 8416 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
213com23 1211 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  ( A  +  B )  <  ( C  +  B )
) )
323expa 1205 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  < 
C  <->  ( A  +  B )  <  ( C  +  B )
) )
43adantrr 479 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  C  <->  ( A  +  B )  <  ( C  +  B ) ) )
5 leadd2 8418 . . . . . 6  |-  ( ( B  e.  RR  /\  D  e.  RR  /\  C  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
653com23 1211 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  D  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
763expb 1206 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
87adantll 476 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D ) ) )
94, 8anbi12d 473 . 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 7967 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
1110adantr 276 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  B
)  e.  RR )
12 readdcl 7967 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1312ancoms 268 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B
)  e.  RR )
1413ad2ant2lr 510 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  B
)  e.  RR )
15 readdcl 7967 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  e.  RR )
1615adantl 277 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  D
)  e.  RR )
17 ltletr 8077 . . 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 1249 . 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 150 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    <-> wb 105    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   RRcr 7840    + caddc 7844    < clt 8022    <_ cle 8023
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-pre-ltwlin 7954  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-iota 5196  df-fv 5243  df-ov 5899  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028
This theorem is referenced by:  leltadd  8434  addgtge0  8437  ltleaddd  8552
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