Theorem max0addsup 11929

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 8290	. . . . . 6 |- 0 e. RR
2		maxabs 11919	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
3	1, 2	mpan2 425	. . . . 5 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) )
4		recn 8276	. . . . . . . 8 |- ( A e. RR -> A e. CC )
5	4	addridd 8438	. . . . . . 7 |- ( A e. RR -> ( A + 0 ) = A )
6	4	subid1d 8589	. . . . . . . 8 |- ( A e. RR -> ( A - 0 ) = A )
7	6	fveq2d 5679	. . . . . . 7 |- ( A e. RR -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
8	5, 7	oveq12d 6076	. . . . . 6 |- ( A e. RR -> ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) = ( A + ( abs ` A ) ) )
9	8	oveq1d 6073	. . . . 5 |- ( A e. RR -> ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) = ( ( A + ( abs ` A ) ) / 2 ) )
10	3, 9	eqtrd 2267	. . . 4 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( A + ( abs ` A ) ) / 2 ) )
11		renegcl 8550	. . . . . 6 |- ( A e. RR -> -u A e. RR )
12		maxabs 11919	. . . . . 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 413	. . . . 5 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) )
14	11	recnd 8318	. . . . . . . 8 |- ( A e. RR -> -u A e. CC )
15	14	addridd 8438	. . . . . . 7 |- ( A e. RR -> ( -u A + 0 ) = -u A )
16	14	subid1d 8589	. . . . . . . . 9 |- ( A e. RR -> ( -u A - 0 ) = -u A )
17	16	fveq2d 5679	. . . . . . . 8 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` -u A ) )
18	4	absnegd 11899	. . . . . . . 8 |- ( A e. RR -> ( abs ` -u A ) = ( abs ` A ) )
19	17, 18	eqtrd 2267	. . . . . . 7 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` A ) )
20	15, 19	oveq12d 6076	. . . . . 6 |- ( A e. RR -> ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) = ( -u A + ( abs ` A ) ) )
21	20	oveq1d 6073	. . . . 5 |- ( A e. RR -> ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
22	13, 21	eqtrd 2267	. . . 4 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
23	10, 22	oveq12d 6076	. . 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 11891	. . . . . 6 |- ( A e. RR -> ( abs ` A ) e. RR )
25	24	recnd 8318	. . . . 5 |- ( A e. RR -> ( abs ` A ) e. CC )
26	4, 25	addcld 8309	. . . 4 |- ( A e. RR -> ( A + ( abs ` A ) ) e. CC )
27	14, 25	addcld 8309	. . . 4 |- ( A e. RR -> ( -u A + ( abs ` A ) ) e. CC )
28		2cnd 9327	. . . 4 |- ( A e. RR -> 2 e. CC )
29		2ap0 9347	. . . . 5 |- 2 # 0
30	29	a1i 9	. . . 4 |- ( A e. RR -> 2 # 0 )
31	26, 27, 28, 30	divdirapd 9120	. . 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 8458	. . . . 5 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) )
33	4	negidd 8590	. . . . . 6 |- ( A e. RR -> ( A + -u A ) = 0 )
34	33	oveq1d 6073	. . . . 5 |- ( A e. RR -> ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) = ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) )
35	25, 25	addcld 8309	. . . . . 6 |- ( A e. RR -> ( ( abs ` A ) + ( abs ` A ) ) e. CC )
36	35	addlidd 8439	. . . . 5 |- ( A e. RR -> ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
37	32, 34, 36	3eqtrd 2271	. . . 4 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
38	37	oveq1d 6073	. . 3 |- ( A e. RR -> ( ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
39	23, 31, 38	3eqtr2d 2273	. 2 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
40	25	2timesd 9498	. . 3 |- ( A e. RR -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
41	40	oveq1d 6073	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
42	25, 28, 30	divcanap3d 9086	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( abs ` A ) )
43	39, 41, 42	3eqtr2d 2273	1 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( abs ` A ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {cpr 3695   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   supcsup 7286   RRcr 8142   0cc0 8143   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   -ucneg 8461   # cap 8872   / cdiv 8963   2c2 9305   abscabs 11707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709

This theorem is referenced by: (None)