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Theorem lesub2 8444
Description: Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
lesub2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  -  B )  <_  ( C  -  A )
) )

Proof of Theorem lesub2
StepHypRef Expression
1 leadd2 8418 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B )
) )
2 simp3 1001 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 simp1 999 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
42, 3readdcld 8017 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
5 simp2 1000 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
6 lesubadd 8421 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
74, 5, 2, 6syl3anc 1249 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
82recnd 8016 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
93recnd 8016 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
105recnd 8016 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
118, 9, 10addsubd 8319 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  -  B )  =  ( ( C  -  B )  +  A ) )
1211breq1d 4028 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( ( C  -  B )  +  A )  <_  C
) )
131, 7, 123bitr2d 216 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( ( C  -  B )  +  A )  <_  C
) )
142, 5resubcld 8368 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  -  B )  e.  RR )
15 leaddsub 8425 . . 3  |-  ( ( ( C  -  B
)  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1614, 3, 2, 15syl3anc 1249 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1713, 16bitrd 188 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  -  B )  <_  ( C  -  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   RRcr 7840    + caddc 7844    <_ cle 8023    - 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-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-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161
This theorem is referenced by:  le2sub  8448  leneg  8452  lesub0  8466  lesub2d  8540
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