Theorem max0addsup 11985

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 8326	. . . . . 6 |- 0 e. RR
2		maxabs 11975	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
3	1, 2	mpan2 429	. . . . 5 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
4		recn 8312	. . . . . . . 8 |- ( A e. RR -> A e. CC )
5	4	addridd 8475	. . . . . . 7 |- ( A e. RR -> ( A + 0 ) = A )
6	4	subid1d 8626	. . . . . . . 8 |- ( A e. RR -> ( A - 0 ) = A )
7	6	fveq2d 5699	. . . . . . 7 |- ( A e. RR -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
8	5, 7	oveq12d 6103	. . . . . 6 |- ( A e. RR -> ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) = ( A + ( abs ` A ) ) )
9	8	oveq1d 6100	. . . . 5 |- ( A e. RR -> ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) = ( ( A + ( abs ` A ) ) / 2 ) )
10	3, 9	eqtrd 2271	. . . 4 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( A + ( abs ` A ) ) / 2 ) )
11		renegcl 8587	. . . . . 6 |- ( A e. RR -> -u A e. RR )
12		maxabs 11975	. . . . . 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 417	. . . . 5 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) )
14	11	recnd 8354	. . . . . . . 8 |- ( A e. RR -> -u A e. CC )
15	14	addridd 8475	. . . . . . 7 |- ( A e. RR -> ( -u A + 0 ) = -u A )
16	14	subid1d 8626	. . . . . . . . 9 |- ( A e. RR -> ( -u A - 0 ) = -u A )
17	16	fveq2d 5699	. . . . . . . 8 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` -u A ) )
18	4	absnegd 11955	. . . . . . . 8 |- ( A e. RR -> ( abs ` -u A ) = ( abs ` A ) )
19	17, 18	eqtrd 2271	. . . . . . 7 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` A ) )
20	15, 19	oveq12d 6103	. . . . . 6 |- ( A e. RR -> ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) = ( -u A + ( abs ` A ) ) )
21	20	oveq1d 6100	. . . . 5 |- ( A e. RR -> ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
22	13, 21	eqtrd 2271	. . . 4 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
23	10, 22	oveq12d 6103	. . 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 11947	. . . . . 6 |- ( A e. RR -> ( abs ` A ) e. RR )
25	24	recnd 8354	. . . . 5 |- ( A e. RR -> ( abs ` A ) e. CC )
26	4, 25	addcld 8345	. . . 4 |- ( A e. RR -> ( A + ( abs ` A ) ) e. CC )
27	14, 25	addcld 8345	. . . 4 |- ( A e. RR -> ( -u A + ( abs ` A ) ) e. CC )
28		2cnd 9377	. . . 4 |- ( A e. RR -> 2 e. CC )
29		2ap0 9397	. . . . 5 |- 2 # 0
30	29	a1i 9	. . . 4 |- ( A e. RR -> 2 # 0 )
31	26, 27, 28, 30	divdirapd 9159	. . 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 8495	. . . . 5 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) )
33	4	negidd 8627	. . . . . 6 |- ( A e. RR -> ( A + -u A ) = 0 )
34	33	oveq1d 6100	. . . . 5 |- ( A e. RR -> ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) = ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) )
35	25, 25	addcld 8345	. . . . . 6 |- ( A e. RR -> ( ( abs ` A ) + ( abs ` A ) ) e. CC )
36	35	addlidd 8476	. . . . 5 |- ( A e. RR -> ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
37	32, 34, 36	3eqtrd 2275	. . . 4 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
38	37	oveq1d 6100	. . 3 |- ( A e. RR -> ( ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
39	23, 31, 38	3eqtr2d 2277	. 2 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
40	25	2timesd 9548	. . 3 |- ( A e. RR -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
41	40	oveq1d 6100	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
42	25, 28, 30	divcanap3d 9125	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( abs ` A ) )
43	39, 41, 42	3eqtr2d 2277	1 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( abs ` A ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   supcsup 7322   RRcr 8178   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   -ucneg 8498   # cap 8909   / cdiv 9002   2c2 9355   abscabs 11763

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765

This theorem is used by: (None)