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Theorem lt2sub 8447
Description: Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
Assertion
Ref Expression
lt2sub  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  D  <  B )  ->  ( A  -  B )  <  ( C  -  D )
) )

Proof of Theorem lt2sub
StepHypRef Expression
1 simpll 527 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
2 simprl 529 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
3 simplr 528 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
4 ltsub1 8445 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <  C  <->  ( A  -  B )  <  ( C  -  B )
) )
51, 2, 3, 4syl3anc 1249 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  C  <->  ( A  -  B )  <  ( C  -  B ) ) )
6 simprr 531 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
7 ltsub2 8446 . . . 4  |-  ( ( D  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( D  <  B  <->  ( C  -  B )  <  ( C  -  D )
) )
86, 3, 2, 7syl3anc 1249 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  <  B  <->  ( C  -  B )  <  ( C  -  D ) ) )
95, 8anbi12d 473 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  D  <  B )  <->  ( ( A  -  B )  < 
( C  -  B
)  /\  ( C  -  B )  <  ( C  -  D )
) ) )
10 resubcl 8251 . . . 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 )
122, 3resubcld 8368 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  B
)  e.  RR )
13 resubcl 8251 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  -  D
)  e.  RR )
1413adantl 277 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  D
)  e.  RR )
15 lttr 8061 . . 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
) ) )
1611, 12, 14, 15syl3anc 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 )
) )
179, 16sylbid 150 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  D  <  B )  ->  ( 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    < clt 8022    - cmin 8158
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-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-lttrn 7955  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-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  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-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161
This theorem is referenced by:  lt2subd  8555
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