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Theorem le2sub 9483
Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
Assertion
Ref Expression
le2sub  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  D  <_  B
)  ->  ( A  -  B )  <_  ( C  -  D )
) )

Proof of Theorem le2sub
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
2 simprl 733 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
3 simplr 732 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
4 lesub1 9478 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <_  C  <->  ( A  -  B )  <_  ( C  -  B )
) )
51, 2, 3, 4syl3anc 1184 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <_  C  <->  ( A  -  B )  <_  ( C  -  B ) ) )
6 simprr 734 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
7 lesub2 9479 . . . 4  |-  ( ( D  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( D  <_  B  <->  ( C  -  B )  <_  ( C  -  D )
) )
86, 3, 2, 7syl3anc 1184 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  <_  B  <->  ( C  -  B )  <_  ( C  -  D ) ) )
95, 8anbi12d 692 . 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 9321 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
1110adantr 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  -  B
)  e.  RR )
122, 3resubcld 9421 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  B
)  e.  RR )
13 resubcl 9321 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  -  D
)  e.  RR )
1413adantl 453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  D
)  e.  RR )
15 letr 9123 . . 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 1184 . 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 207 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    <-> wb 177    /\ wa 359    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RRcr 8945    <_ cle 9077    - cmin 9247
This theorem is referenced by:  le2subd  9601  fsumharmonic  20803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250
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