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Theorem 2wsms 25711
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 8843 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  CC )
2 recn 8843 . . . . . . . . . . . . 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 9067 . . . . . . . . . . 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 11934 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  RR )
87recnd 8877 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  CC )
9 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
109a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  e.  CC )
11 2ne0 9845 . . . . . . . . 9  |-  2  =/=  0
1211a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  2  =/=  0 )
138, 10, 12divcan2d 9554 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( abs `  ( A  -  B ) )  /  2 ) )  =  ( abs `  ( A  -  B )
) )
14 resubcl 9127 . . . . . . . . . . 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 8877 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  -  B )  e.  CC )
1716abscld 11934 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  RR )
1817recnd 8877 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( abs `  ( A  -  B ) )  e.  CC )
1913, 18eqeltrd 2370 . . . . . 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 8840 . . . . . . 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 2301 . . . . . 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 8870 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  A )  e.  CC )
2625, 21addcomd 9030 . . . . 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 8877 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
29282timesd 9970 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  B )  =  ( B  +  B ) )
3029oveq1d 5889 . . . . . . 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 9073 . . . . . . . . 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 8926 . . . . . . . . . . . 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 11828 . . . . . . . . . . 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 2329 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  =  ( 2  x.  ( ( abs `  ( A  -  B )
)  /  2 ) ) )
4021, 20, 19subaddd 9191 . . . . . . . . 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 9030 . . . . . . . 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 2334 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  +  B
)  -  B )  =  ( ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  +  A
) )
4430, 43eqtrd 2328 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  B
)  -  B )  =  ( ( 2  x.  ( ( abs `  ( A  -  B
) )  /  2
) )  +  A
) )
45 mulcl 8837 . . . . . . . 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 9191 . . . . . 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 2332 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( 2  x.  (
( abs `  ( A  -  B )
)  /  2 ) )  +  ( A  +  B ) )  =  ( 2  x.  B ) )
50 addcl 8835 . . . . . 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 9191 . . . 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 25707 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( A  -  B )
)  /  2 )  e.  RR+ )
5554rpcnd 10408 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( abs `  ( A  -  B )
)  /  2 )  e.  CC )
5610, 21, 55subdid 9251 . . 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 9554 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( A  +  B )  /  2 ) )  =  ( A  +  B ) )
5853, 56, 573eqtr4rd 2339 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
2  x.  ( ( A  +  B )  /  2 ) )  =  ( 2  x.  ( B  -  (
( abs `  ( A  -  B )
)  /  2 ) ) ) )
59 halfaddsubcl 9960 . . . . 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 25708 . . . . 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 8877 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  ( ( abs `  ( A  -  B ) )  / 
2 ) )  e.  CC )
6561, 64, 10, 12mulcand 9417 . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   abscabs 11735
This theorem is referenced by:  msra3  25712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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