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Theorem lesub2 8369
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 8343 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B )
) )
2 simp3 994 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 simp1 992 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
42, 3readdcld 7942 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
5 simp2 993 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
6 lesubadd 8346 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
74, 5, 2, 6syl3anc 1233 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( C  +  A )  <_  ( C  +  B )
) )
82recnd 7941 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
93recnd 7941 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
105recnd 7941 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
118, 9, 10addsubd 8244 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  -  B )  =  ( ( C  -  B )  +  A ) )
1211breq1d 3997 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  +  A )  -  B
)  <_  C  <->  ( ( C  -  B )  +  A )  <_  C
) )
131, 7, 123bitr2d 215 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( ( C  -  B )  +  A )  <_  C
) )
142, 5resubcld 8293 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  -  B )  e.  RR )
15 leaddsub 8350 . . 3  |-  ( ( ( C  -  B
)  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1614, 3, 2, 15syl3anc 1233 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( C  -  B )  +  A
)  <_  C  <->  ( C  -  B )  <_  ( C  -  A )
) )
1713, 16bitrd 187 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 104    /\ w3a 973    e. wcel 2141   class class class wbr 3987  (class class class)co 5851   RRcr 7766    + caddc 7770    <_ cle 7948    - cmin 8083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltadd 7883
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086
This theorem is referenced by:  le2sub  8373  leneg  8377  lesub0  8391  lesub2d  8465
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