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Theorem le2sub 9360
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 730 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
2 simprl 732 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
3 simplr 731 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
4 lesub1 9355 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <_  C  <->  ( A  -  B )  <_  ( C  -  B )
) )
51, 2, 3, 4syl3anc 1182 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <_  C  <->  ( A  -  B )  <_  ( C  -  B ) ) )
6 simprr 733 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
7 lesub2 9356 . . . 4  |-  ( ( D  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( D  <_  B  <->  ( C  -  B )  <_  ( C  -  D )
) )
86, 3, 2, 7syl3anc 1182 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  <_  B  <->  ( C  -  B )  <_  ( C  -  D ) ) )
95, 8anbi12d 691 . 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 9198 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
1110adantr 451 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  -  B
)  e.  RR )
122, 3resubcld 9298 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  B
)  e.  RR )
13 resubcl 9198 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  -  D
)  e.  RR )
1413adantl 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  -  D
)  e.  RR )
15 letr 9001 . . 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 1182 . 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 206 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 176    /\ wa 358    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   RRcr 8823    <_ cle 8955    - cmin 9124
This theorem is referenced by:  le2subd  9478  fsumharmonic  20411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127
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