Theorem max0addsup 11161

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

Step	Hyp	Ref	Expression
1		0re 7899	. . . . . 6 |- 0 e. RR
2		maxabs 11151	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
3	1, 2	mpan2 422	. . . . 5 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
4		recn 7886	. . . . . . . 8 |- ( A e. RR -> A e. CC )
5	4	addid1d 8047	. . . . . . 7 |- ( A e. RR -> ( A + 0 ) = A )
6	4	subid1d 8198	. . . . . . . 8 |- ( A e. RR -> ( A - 0 ) = A )
7	6	fveq2d 5490	. . . . . . 7 |- ( A e. RR -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
8	5, 7	oveq12d 5860	. . . . . 6 |- ( A e. RR -> ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) = ( A + ( abs ` A ) ) )
9	8	oveq1d 5857	. . . . 5 |- ( A e. RR -> ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) = ( ( A + ( abs ` A ) ) / 2 ) )
10	3, 9	eqtrd 2198	. . . 4 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( A + ( abs ` A ) ) / 2 ) )
11		renegcl 8159	. . . . . 6 |- ( A e. RR -> -u A e. RR )
12		maxabs 11151	. . . . . 6 |- ( ( -u A e. RR /\ 0 e. RR ) -> sup ( { -u A , 0 } , RR , < ) = ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) )
13	11, 1, 12	sylancl 410	. . . . 5 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) )
14	11	recnd 7927	. . . . . . . 8 |- ( A e. RR -> -u A e. CC )
15	14	addid1d 8047	. . . . . . 7 |- ( A e. RR -> ( -u A + 0 ) = -u A )
16	14	subid1d 8198	. . . . . . . . 9 |- ( A e. RR -> ( -u A - 0 ) = -u A )
17	16	fveq2d 5490	. . . . . . . 8 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` -u A ) )
18	4	absnegd 11131	. . . . . . . 8 |- ( A e. RR -> ( abs ` -u A ) = ( abs ` A ) )
19	17, 18	eqtrd 2198	. . . . . . 7 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` A ) )
20	15, 19	oveq12d 5860	. . . . . 6 |- ( A e. RR -> ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) = ( -u A + ( abs ` A ) ) )
21	20	oveq1d 5857	. . . . 5 |- ( A e. RR -> ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
22	13, 21	eqtrd 2198	. . . 4 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
23	10, 22	oveq12d 5860	. . 3 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( ( ( A + ( abs ` A ) ) / 2 ) + ( ( -u A + ( abs ` A ) ) / 2 ) ) )
24	4	abscld 11123	. . . . . 6 |- ( A e. RR -> ( abs ` A ) e. RR )
25	24	recnd 7927	. . . . 5 |- ( A e. RR -> ( abs ` A ) e. CC )
26	4, 25	addcld 7918	. . . 4 |- ( A e. RR -> ( A + ( abs ` A ) ) e. CC )
27	14, 25	addcld 7918	. . . 4 |- ( A e. RR -> ( -u A + ( abs ` A ) ) e. CC )
28		2cnd 8930	. . . 4 |- ( A e. RR -> 2 e. CC )
29		2ap0 8950	. . . . 5 |- 2 # 0
30	29	a1i 9	. . . 4 |- ( A e. RR -> 2 # 0 )
31	26, 27, 28, 30	divdirapd 8725	. . 3 |- ( A e. RR -> ( ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) / 2 ) = ( ( ( A + ( abs ` A ) ) / 2 ) + ( ( -u A + ( abs ` A ) ) / 2 ) ) )
32	4, 25, 14, 25	add4d 8067	. . . . 5 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) )
33	4	negidd 8199	. . . . . 6 |- ( A e. RR -> ( A + -u A ) = 0 )
34	33	oveq1d 5857	. . . . 5 |- ( A e. RR -> ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) = ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) )
35	25, 25	addcld 7918	. . . . . 6 |- ( A e. RR -> ( ( abs ` A ) + ( abs ` A ) ) e. CC )
36	35	addid2d 8048	. . . . 5 |- ( A e. RR -> ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
37	32, 34, 36	3eqtrd 2202	. . . 4 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
38	37	oveq1d 5857	. . 3 |- ( A e. RR -> ( ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
39	23, 31, 38	3eqtr2d 2204	. 2 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
40	25	2timesd 9099	. . 3 |- ( A e. RR -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
41	40	oveq1d 5857	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
42	25, 28, 30	divcanap3d 8691	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( abs ` A ) )
43	39, 41, 42	3eqtr2d 2204	1 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( abs ` A ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   {cpr 3577   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   supcsup 6947   RRcr 7752   0cc0 7753   + caddc 7756   x. cmul 7758   < clt 7933   - cmin 8069   -ucneg 8070   # cap 8479   / cdiv 8568   2c2 8908   abscabs 10939

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by: (None)