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Theorem lesub2 7452
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 7426 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B )
) )
2 simp3 906 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 simp1 904 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
42, 3readdcld 7055 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
5 simp2 905 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
6 lesubadd 7429 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
74, 5, 2, 6syl3anc 1135 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
82recnd 7054 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
93recnd 7054 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
105recnd 7054 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
118, 9, 10addsubd 7343 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  -  B )  =  ( ( C  -  B )  +  A ) )
1211breq1d 3774 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( ( C  -  B )  +  A )  <_  C
) )
131, 7, 123bitr2d 205 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( ( C  -  B )  +  A )  <_  C
) )
142, 5resubcld 7379 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  -  B )  e.  RR )
15 leaddsub 7433 . . 3  |-  ( ( ( C  -  B
)  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1614, 3, 2, 15syl3anc 1135 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1713, 16bitrd 177 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 98    /\ w3a 885    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888    + caddc 6892    <_ cle 7061    - cmin 7182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185
This theorem is referenced by:  le2sub  7456  leneg  7460  lesub0  7474  lesub2d  7544
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