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Theorem max0addsup 10998
Description: The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
Assertion
Ref Expression
max0addsup  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( abs `  A
) )

Proof of Theorem max0addsup
StepHypRef Expression
1 0re 7773 . . . . . 6  |-  0  e.  RR
2 maxabs 10988 . . . . . 6  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( ( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  /  2
) )
31, 2mpan2 421 . . . . 5  |-  ( A  e.  RR  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( ( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  / 
2 ) )
4 recn 7760 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
54addid1d 7918 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  0 )  =  A )
64subid1d 8069 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  0 )  =  A )
76fveq2d 5425 . . . . . . 7  |-  ( A  e.  RR  ->  ( abs `  ( A  - 
0 ) )  =  ( abs `  A
) )
85, 7oveq12d 5792 . . . . . 6  |-  ( A  e.  RR  ->  (
( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  =  ( A  +  ( abs `  A ) ) )
98oveq1d 5789 . . . . 5  |-  ( A  e.  RR  ->  (
( ( A  + 
0 )  +  ( abs `  ( A  -  0 ) ) )  /  2 )  =  ( ( A  +  ( abs `  A
) )  /  2
) )
103, 9eqtrd 2172 . . . 4  |-  ( A  e.  RR  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( A  +  ( abs `  A ) )  / 
2 ) )
11 renegcl 8030 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
12 maxabs 10988 . . . . . 6  |-  ( (
-u A  e.  RR  /\  0  e.  RR )  ->  sup ( { -u A ,  0 } ,  RR ,  <  )  =  ( ( (
-u A  +  0 )  +  ( abs `  ( -u A  - 
0 ) ) )  /  2 ) )
1311, 1, 12sylancl 409 . . . . 5  |-  ( A  e.  RR  ->  sup ( { -u A , 
0 } ,  RR ,  <  )  =  ( ( ( -u A  +  0 )  +  ( abs `  ( -u A  -  0 ) ) )  /  2
) )
1411recnd 7801 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  CC )
1514addid1d 7918 . . . . . . 7  |-  ( A  e.  RR  ->  ( -u A  +  0 )  =  -u A )
1614subid1d 8069 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( -u A  -  0 )  =  -u A )
1716fveq2d 5425 . . . . . . . 8  |-  ( A  e.  RR  ->  ( abs `  ( -u A  -  0 ) )  =  ( abs `  -u A
) )
184absnegd 10968 . . . . . . . 8  |-  ( A  e.  RR  ->  ( abs `  -u A )  =  ( abs `  A
) )
1917, 18eqtrd 2172 . . . . . . 7  |-  ( A  e.  RR  ->  ( abs `  ( -u A  -  0 ) )  =  ( abs `  A
) )
2015, 19oveq12d 5792 . . . . . 6  |-  ( A  e.  RR  ->  (
( -u A  +  0 )  +  ( abs `  ( -u A  - 
0 ) ) )  =  ( -u A  +  ( abs `  A
) ) )
2120oveq1d 5789 . . . . 5  |-  ( A  e.  RR  ->  (
( ( -u A  +  0 )  +  ( abs `  ( -u A  -  0 ) ) )  /  2
)  =  ( (
-u A  +  ( abs `  A ) )  /  2 ) )
2213, 21eqtrd 2172 . . . 4  |-  ( A  e.  RR  ->  sup ( { -u A , 
0 } ,  RR ,  <  )  =  ( ( -u A  +  ( abs `  A ) )  /  2 ) )
2310, 22oveq12d 5792 . . 3  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( ( ( A  +  ( abs `  A
) )  /  2
)  +  ( (
-u A  +  ( abs `  A ) )  /  2 ) ) )
244abscld 10960 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
2524recnd 7801 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  A )  e.  CC )
264, 25addcld 7792 . . . 4  |-  ( A  e.  RR  ->  ( A  +  ( abs `  A ) )  e.  CC )
2714, 25addcld 7792 . . . 4  |-  ( A  e.  RR  ->  ( -u A  +  ( abs `  A ) )  e.  CC )
28 2cnd 8800 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
29 2ap0 8820 . . . . 5  |-  2 #  0
3029a1i 9 . . . 4  |-  ( A  e.  RR  ->  2 #  0 )
3126, 27, 28, 30divdirapd 8596 . . 3  |-  ( A  e.  RR  ->  (
( ( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  /  2 )  =  ( ( ( A  +  ( abs `  A
) )  /  2
)  +  ( (
-u A  +  ( abs `  A ) )  /  2 ) ) )
324, 25, 14, 25add4d 7938 . . . . 5  |-  ( A  e.  RR  ->  (
( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  =  ( ( A  +  -u A )  +  ( ( abs `  A
)  +  ( abs `  A ) ) ) )
334negidd 8070 . . . . . 6  |-  ( A  e.  RR  ->  ( A  +  -u A )  =  0 )
3433oveq1d 5789 . . . . 5  |-  ( A  e.  RR  ->  (
( A  +  -u A )  +  ( ( abs `  A
)  +  ( abs `  A ) ) )  =  ( 0  +  ( ( abs `  A
)  +  ( abs `  A ) ) ) )
3525, 25addcld 7792 . . . . . 6  |-  ( A  e.  RR  ->  (
( abs `  A
)  +  ( abs `  A ) )  e.  CC )
3635addid2d 7919 . . . . 5  |-  ( A  e.  RR  ->  (
0  +  ( ( abs `  A )  +  ( abs `  A
) ) )  =  ( ( abs `  A
)  +  ( abs `  A ) ) )
3732, 34, 363eqtrd 2176 . . . 4  |-  ( A  e.  RR  ->  (
( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  =  ( ( abs `  A )  +  ( abs `  A ) ) )
3837oveq1d 5789 . . 3  |-  ( A  e.  RR  ->  (
( ( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  /  2 )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
3923, 31, 383eqtr2d 2178 . 2  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
40252timesd 8969 . . 3  |-  ( A  e.  RR  ->  (
2  x.  ( abs `  A ) )  =  ( ( abs `  A
)  +  ( abs `  A ) ) )
4140oveq1d 5789 . 2  |-  ( A  e.  RR  ->  (
( 2  x.  ( abs `  A ) )  /  2 )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
4225, 28, 30divcanap3d 8562 . 2  |-  ( A  e.  RR  ->  (
( 2  x.  ( abs `  A ) )  /  2 )  =  ( abs `  A
) )
4339, 41, 423eqtr2d 2178 1  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( abs `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   {cpr 3528   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   supcsup 6869   RRcr 7626   0cc0 7627    + caddc 7630    x. cmul 7632    < clt 7807    - cmin 7940   -ucneg 7941   # cap 8350    / cdiv 8439   2c2 8778   abscabs 10776
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778
This theorem is referenced by: (None)
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