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Theorem max0addsup 10493
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 7409 . . . . . 6  |-  0  e.  RR
2 maxabs 10483 . . . . . 6  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( ( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  /  2
) )
31, 2mpan2 416 . . . . 5  |-  ( A  e.  RR  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( ( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  / 
2 ) )
4 recn 7396 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
54addid1d 7552 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  0 )  =  A )
64subid1d 7703 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  0 )  =  A )
76fveq2d 5260 . . . . . . 7  |-  ( A  e.  RR  ->  ( abs `  ( A  - 
0 ) )  =  ( abs `  A
) )
85, 7oveq12d 5612 . . . . . 6  |-  ( A  e.  RR  ->  (
( A  +  0 )  +  ( abs `  ( A  -  0 ) ) )  =  ( A  +  ( abs `  A ) ) )
98oveq1d 5609 . . . . 5  |-  ( A  e.  RR  ->  (
( ( A  + 
0 )  +  ( abs `  ( A  -  0 ) ) )  /  2 )  =  ( ( A  +  ( abs `  A
) )  /  2
) )
103, 9eqtrd 2117 . . . 4  |-  ( A  e.  RR  ->  sup ( { A ,  0 } ,  RR ,  <  )  =  ( ( A  +  ( abs `  A ) )  / 
2 ) )
11 renegcl 7664 . . . . . 6  |-  ( A  e.  RR  ->  -u A  e.  RR )
12 maxabs 10483 . . . . . 6  |-  ( (
-u A  e.  RR  /\  0  e.  RR )  ->  sup ( { -u A ,  0 } ,  RR ,  <  )  =  ( ( (
-u A  +  0 )  +  ( abs `  ( -u A  - 
0 ) ) )  /  2 ) )
1311, 1, 12sylancl 404 . . . . 5  |-  ( A  e.  RR  ->  sup ( { -u A , 
0 } ,  RR ,  <  )  =  ( ( ( -u A  +  0 )  +  ( abs `  ( -u A  -  0 ) ) )  /  2
) )
1411recnd 7437 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  CC )
1514addid1d 7552 . . . . . . 7  |-  ( A  e.  RR  ->  ( -u A  +  0 )  =  -u A )
1614subid1d 7703 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( -u A  -  0 )  =  -u A )
1716fveq2d 5260 . . . . . . . 8  |-  ( A  e.  RR  ->  ( abs `  ( -u A  -  0 ) )  =  ( abs `  -u A
) )
184absnegd 10463 . . . . . . . 8  |-  ( A  e.  RR  ->  ( abs `  -u A )  =  ( abs `  A
) )
1917, 18eqtrd 2117 . . . . . . 7  |-  ( A  e.  RR  ->  ( abs `  ( -u A  -  0 ) )  =  ( abs `  A
) )
2015, 19oveq12d 5612 . . . . . 6  |-  ( A  e.  RR  ->  (
( -u A  +  0 )  +  ( abs `  ( -u A  - 
0 ) ) )  =  ( -u A  +  ( abs `  A
) ) )
2120oveq1d 5609 . . . . 5  |-  ( A  e.  RR  ->  (
( ( -u A  +  0 )  +  ( abs `  ( -u A  -  0 ) ) )  /  2
)  =  ( (
-u A  +  ( abs `  A ) )  /  2 ) )
2213, 21eqtrd 2117 . . . 4  |-  ( A  e.  RR  ->  sup ( { -u A , 
0 } ,  RR ,  <  )  =  ( ( -u A  +  ( abs `  A ) )  /  2 ) )
2310, 22oveq12d 5612 . . 3  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( ( ( A  +  ( abs `  A
) )  /  2
)  +  ( (
-u A  +  ( abs `  A ) )  /  2 ) ) )
244abscld 10455 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
2524recnd 7437 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  A )  e.  CC )
264, 25addcld 7428 . . . 4  |-  ( A  e.  RR  ->  ( A  +  ( abs `  A ) )  e.  CC )
2714, 25addcld 7428 . . . 4  |-  ( A  e.  RR  ->  ( -u A  +  ( abs `  A ) )  e.  CC )
28 2cnd 8407 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
29 2ap0 8427 . . . . 5  |-  2 #  0
3029a1i 9 . . . 4  |-  ( A  e.  RR  ->  2 #  0 )
3126, 27, 28, 30divdirapd 8210 . . 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 7572 . . . . 5  |-  ( A  e.  RR  ->  (
( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  =  ( ( A  +  -u A )  +  ( ( abs `  A
)  +  ( abs `  A ) ) ) )
334negidd 7704 . . . . . 6  |-  ( A  e.  RR  ->  ( A  +  -u A )  =  0 )
3433oveq1d 5609 . . . . 5  |-  ( A  e.  RR  ->  (
( A  +  -u A )  +  ( ( abs `  A
)  +  ( abs `  A ) ) )  =  ( 0  +  ( ( abs `  A
)  +  ( abs `  A ) ) ) )
3525, 25addcld 7428 . . . . . 6  |-  ( A  e.  RR  ->  (
( abs `  A
)  +  ( abs `  A ) )  e.  CC )
3635addid2d 7553 . . . . 5  |-  ( A  e.  RR  ->  (
0  +  ( ( abs `  A )  +  ( abs `  A
) ) )  =  ( ( abs `  A
)  +  ( abs `  A ) ) )
3732, 34, 363eqtrd 2121 . . . 4  |-  ( A  e.  RR  ->  (
( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  =  ( ( abs `  A )  +  ( abs `  A ) ) )
3837oveq1d 5609 . . 3  |-  ( A  e.  RR  ->  (
( ( A  +  ( abs `  A ) )  +  ( -u A  +  ( abs `  A ) ) )  /  2 )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
3923, 31, 383eqtr2d 2123 . 2  |-  ( A  e.  RR  ->  ( sup ( { A , 
0 } ,  RR ,  <  )  +  sup ( { -u A , 
0 } ,  RR ,  <  ) )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
40252timesd 8568 . . 3  |-  ( A  e.  RR  ->  (
2  x.  ( abs `  A ) )  =  ( ( abs `  A
)  +  ( abs `  A ) ) )
4140oveq1d 5609 . 2  |-  ( A  e.  RR  ->  (
( 2  x.  ( abs `  A ) )  /  2 )  =  ( ( ( abs `  A )  +  ( abs `  A ) )  /  2 ) )
4225, 28, 30divcanap3d 8177 . 2  |-  ( A  e.  RR  ->  (
( 2  x.  ( abs `  A ) )  /  2 )  =  ( abs `  A
) )
4339, 41, 423eqtr2d 2123 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 1287    e. wcel 1436   {cpr 3426   class class class wbr 3814   ` cfv 4972  (class class class)co 5594   supcsup 6598   RRcr 7270   0cc0 7271    + caddc 7274    x. cmul 7276    < clt 7443    - cmin 7574   -ucneg 7575   # cap 7976    / cdiv 8055   2c2 8384   abscabs 10271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-rp 9044  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273
This theorem is referenced by: (None)
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