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Theorem avgle2 10133
Description: Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
Assertion
Ref Expression
avgle2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( ( A  +  B
)  /  2 )  <_  B ) )

Proof of Theorem avgle2
StepHypRef Expression
1 avglt1 10130 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B  <  ( ( B  +  A )  / 
2 ) ) )
21ancoms 440 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  ( ( B  +  A )  / 
2 ) ) )
3 recn 9006 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9006 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 9177 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
63, 4, 5syl2an 464 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
76oveq1d 6028 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  /  2
)  =  ( ( B  +  A )  /  2 ) )
87breq2d 4158 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  (
( A  +  B
)  /  2 )  <-> 
B  <  ( ( B  +  A )  /  2 ) ) )
92, 8bitr4d 248 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  ( ( A  +  B )  / 
2 ) ) )
109notbid 286 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  B  < 
A  <->  -.  B  <  ( ( A  +  B
)  /  2 ) ) )
11 lenlt 9080 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
12 readdcl 8999 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
13 rehalfcl 10119 . . . 4  |-  ( ( A  +  B )  e.  RR  ->  (
( A  +  B
)  /  2 )  e.  RR )
1412, 13syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  /  2
)  e.  RR )
15 lenlt 9080 . . 3  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <_  B  <->  -.  B  <  ( ( A  +  B )  /  2 ) ) )
1614, 15sylancom 649 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <_  B  <->  -.  B  <  ( ( A  +  B )  /  2 ) ) )
1710, 11, 163bitr4d 277 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( ( A  +  B
)  /  2 )  <_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146  (class class class)co 6013   CCcc 8914   RRcr 8915    + caddc 8919    < clt 9046    <_ cle 9047    / cdiv 9602   2c2 9974
This theorem is referenced by:  avgle  10134  pilem3  20229
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983
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