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Theorem absmax 11813
Description: The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)
Assertion
Ref Expression
absmax  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A  <_  B ,  B ,  A )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )

Proof of Theorem absmax
StepHypRef Expression
1 recn 8827 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
2 2cn 9816 . . . . . . 7  |-  2  e.  CC
3 2ne0 9829 . . . . . . 7  |-  2  =/=  0
4 divcan3 9448 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  A
)  /  2 )  =  A )
52, 3, 4mp3an23 1269 . . . . . 6  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  /  2 )  =  A )
61, 5syl 15 . . . . 5  |-  ( A  e.  RR  ->  (
( 2  x.  A
)  /  2 )  =  A )
76ad2antlr 707 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( (
2  x.  A )  /  2 )  =  A )
8 ltle 8910 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  ->  B  <_  A )
)
98imp 418 . . . . . . . 8  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  B  <_  A )
10 abssubge0 11811 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  B  <_  A )  ->  ( abs `  ( A  -  B ) )  =  ( A  -  B
) )
11103expa 1151 . . . . . . . 8  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <_  A
)  ->  ( abs `  ( A  -  B
) )  =  ( A  -  B ) )
129, 11syldan 456 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( abs `  ( A  -  B
) )  =  ( A  -  B ) )
1312oveq2d 5874 . . . . . 6  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  =  ( ( A  +  B
)  +  ( A  -  B ) ) )
14 recn 8827 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
15 simpr 447 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  A  e.  CC )
16 simpl 443 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  B  e.  CC )
1715, 16, 15ppncand 9197 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( A  +  A ) )
18 2times 9843 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
1918adantl 452 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
2017, 19eqtr4d 2318 . . . . . . . 8  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
2114, 1, 20syl2an 463 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A ) )
2221adantr 451 . . . . . 6  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( ( A  +  B )  +  ( A  -  B ) )  =  ( 2  x.  A
) )
2313, 22eqtrd 2315 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  =  ( 2  x.  A ) )
2423oveq1d 5873 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  =  ( ( 2  x.  A
)  /  2 ) )
25 ltnle 8902 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  -.  A  <_  B )
)
2625biimpa 470 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  -.  A  <_  B )
27 iffalse 3572 . . . . 5  |-  ( -.  A  <_  B  ->  if ( A  <_  B ,  B ,  A )  =  A )
2826, 27syl 15 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  if ( A  <_  B ,  B ,  A )  =  A )
297, 24, 283eqtr4rd 2326 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  B  <  A
)  ->  if ( A  <_  B ,  B ,  A )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
3029ancom1s 780 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  if ( A  <_  B ,  B ,  A )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
31 divcan3 9448 . . . . . 6  |-  ( ( B  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  B
)  /  2 )  =  B )
322, 3, 31mp3an23 1269 . . . . 5  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  /  2 )  =  B )
3314, 32syl 15 . . . 4  |-  ( B  e.  RR  ->  (
( 2  x.  B
)  /  2 )  =  B )
3433ad2antlr 707 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( (
2  x.  B )  /  2 )  =  B )
35 abssuble0 11812 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A
) )
36353expa 1151 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( abs `  ( A  -  B
) )  =  ( B  -  A ) )
3736oveq2d 5874 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  =  ( ( A  +  B
)  +  ( B  -  A ) ) )
38 simpr 447 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
39 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
4038, 39, 38ppncand 9197 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  A )  +  ( B  -  A ) )  =  ( B  +  B ) )
41 addcom 8998 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
4241oveq1d 5873 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( B  -  A ) )  =  ( ( B  +  A )  +  ( B  -  A ) ) )
43 2times 9843 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
4443adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
4540, 42, 443eqtr4d 2325 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( B  -  A ) )  =  ( 2  x.  B ) )
461, 14, 45syl2an 463 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( B  -  A ) )  =  ( 2  x.  B ) )
4746adantr 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( A  +  B )  +  ( B  -  A ) )  =  ( 2  x.  B
) )
4837, 47eqtrd 2315 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  =  ( 2  x.  B ) )
4948oveq1d 5873 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  =  ( ( 2  x.  B
)  /  2 ) )
50 iftrue 3571 . . . 4  |-  ( A  <_  B  ->  if ( A  <_  B ,  B ,  A )  =  B )
5150adantl 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  if ( A  <_  B ,  B ,  A )  =  B )
5234, 49, 513eqtr4rd 2326 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  if ( A  <_  B ,  B ,  A )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
53 simpr 447 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54 simpl 443 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
5530, 52, 53, 54ltlecasei 8928 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A  <_  B ,  B ,  A )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   abscabs 11719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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