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Theorem 2wsms 25608
Description: Two ways to state the midpoint of a segment. (Contributed by FL, 3-Jan-2008.)
Assertion
Ref Expression
2wsms  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A  +  B
)  /  2 )  =  ( B  -  ( ( abs `  ( A  -  B )
)  /  2 ) ) )

Proof of Theorem 2wsms
StepHypRef Expression
1 recn 8827 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  CC )
2 recn 8827 . . . . . . . . . . . . 13  |-  ( B  e.  RR  ->  B  e.  CC )
31, 2anim12i 549 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  CC  /\  B  e.  CC ) )
433adant3 975 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  e.  CC  /\  B  e.  CC ) )
5 subcl 9051 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
64, 5syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  -  B )  e.  CC )
76abscld 11918 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  RR )
87recnd 8861 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  CC )
9 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
109a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  e.  CC )
11 2ne0 9829 . . . . . . . . 9  |-  2  =/=  0
1211a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  =/=  0 )
138, 10, 12divcan2d 9538 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  =  ( abs `  ( A  -  B )
) )
14 resubcl 9111 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
15143adant3 975 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  -  B )  e.  RR )
1615recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  -  B )  e.  CC )
1716abscld 11918 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  RR )
1817recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  CC )
1913, 18eqeltrd 2357 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  e.  CC )
2013ad2ant1 976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
2123ad2ant2 977 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
22 addass 8824 . . . . . . 7  |-  ( ( ( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) )  +  A )  +  B )  =  ( ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) )  +  ( A  +  B ) ) )
2322eqcomd 2288 . . . . . 6  |-  ( ( ( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  ( A  +  B ) )  =  ( ( ( 2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  +  A )  +  B ) )
2419, 20, 21, 23syl3anc 1182 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  ( A  +  B ) )  =  ( ( ( 2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  +  A )  +  B ) )
2519, 20addcld 8854 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  A )  e.  CC )
2625, 21addcomd 9014 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) )  +  A )  +  B )  =  ( B  +  ( ( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  A ) ) )
27 simp2 956 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
2827recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
29282timesd 9954 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  B )  =  ( B  +  B ) )
3029oveq1d 5873 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  B
)  -  B )  =  ( ( B  +  B )  -  B ) )
312, 2jca 518 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  e.  CC  /\  B  e.  CC ) )
32313ad2ant2 977 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  e.  CC  /\  B  e.  CC ) )
33 pncan 9057 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  B )  -  B
)  =  B )
3432, 33syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  +  B
)  -  B )  =  B )
35 ltle 8910 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
36353impia 1148 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <_  B )
37 abssuble0 11812 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A
) )
3836, 37syld3an3 1227 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A
) )
3913, 38eqtr2d 2316 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =  ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) ) )
4021, 20, 19subaddd 9175 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  A
)  =  ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  <->  ( A  +  ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) ) )  =  B ) )
4139, 40mpbid 201 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) ) )  =  B )
4220, 19addcomd 9014 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) ) )  =  ( ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) )  +  A ) )
4334, 41, 423eqtr2d 2321 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  +  B
)  -  B )  =  ( ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  +  A
) )
4430, 43eqtrd 2315 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  B
)  -  B )  =  ( ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  +  A
) )
45 mulcl 8821 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
469, 28, 45sylancr 644 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  B )  e.  CC )
4746, 21, 25subaddd 9175 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  B )  -  B
)  =  ( ( 2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  +  A )  <->  ( B  +  ( ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  +  A
) )  =  ( 2  x.  B ) ) )
4844, 47mpbid 201 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  +  ( (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  +  A ) )  =  ( 2  x.  B ) )
4924, 26, 483eqtrd 2319 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  ( A  +  B ) )  =  ( 2  x.  B ) )
50 addcl 8819 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
514, 50syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  +  B )  e.  CC )
5246, 19, 51subaddd 9175 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( ( 2  x.  B )  -  (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) ) )  =  ( A  +  B )  <->  ( (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  +  ( A  +  B ) )  =  ( 2  x.  B
) ) )
5349, 52mpbird 223 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  B
)  -  ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) ) )  =  ( A  +  B
) )
54 dmse2 25604 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( A  -  B )
)  /  2 )  e.  RR+ )
5554rpcnd 10392 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( A  -  B )
)  /  2 )  e.  CC )
5610, 21, 55subdid 9235 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( B  -  ( ( abs `  ( A  -  B
) )  /  2
) ) )  =  ( ( 2  x.  B )  -  (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) ) ) )
5751, 10, 12divcan2d 9538 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( A  +  B )  /  2 ) )  =  ( A  +  B ) )
5853, 56, 573eqtr4rd 2326 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( A  +  B )  /  2 ) )  =  ( 2  x.  ( B  -  (
( abs `  ( A  -  B )
)  /  2 ) ) ) )
59 halfaddsubcl 9944 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  / 
2 )  e.  CC  /\  ( ( A  -  B )  /  2
)  e.  CC ) )
6059simpld 445 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  /  2
)  e.  CC )
614, 60syl 15 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A  +  B
)  /  2 )  e.  CC )
62 msr3 25605 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  -  (
( abs `  ( A  -  B )
)  /  2 ) )  e.  RR )
63623adant3 975 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  ( ( abs `  ( A  -  B ) )  / 
2 ) )  e.  RR )
6463recnd 8861 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  ( ( abs `  ( A  -  B ) )  / 
2 ) )  e.  CC )
6561, 64, 10, 12mulcand 9401 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( A  +  B
)  /  2 ) )  =  ( 2  x.  ( B  -  ( ( abs `  ( A  -  B )
)  /  2 ) ) )  <->  ( ( A  +  B )  /  2 )  =  ( B  -  (
( abs `  ( A  -  B )
)  /  2 ) ) ) )
6658, 65mpbid 201 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A  +  B
)  /  2 )  =  ( B  -  ( ( abs `  ( A  -  B )
)  /  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   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 is referenced by:  msra3  25609
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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