Theorem max0addsup 11779

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 8178	. . . . . 6 |- 0 e. RR
2		maxabs 11769	. . . . . 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 8164	. . . . . . . 8 |- ( A e. RR -> A e. CC )
5	4	addridd 8327	. . . . . . 7 |- ( A e. RR -> ( A + 0 ) = A )
6	4	subid1d 8478	. . . . . . . 8 |- ( A e. RR -> ( A - 0 ) = A )
7	6	fveq2d 5643	. . . . . . 7 |- ( A e. RR -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
8	5, 7	oveq12d 6035	. . . . . 6 |- ( A e. RR -> ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) = ( A + ( abs ` A ) ) )
9	8	oveq1d 6032	. . . . 5 |- ( A e. RR -> ( ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) / 2 ) = ( ( A + ( abs ` A ) ) / 2 ) )
10	3, 9	eqtrd 2264	. . . 4 |- ( A e. RR -> sup ( { A , 0 } , RR , < ) = ( ( A + ( abs ` A ) ) / 2 ) )
11		renegcl 8439	. . . . . 6 |- ( A e. RR -> -u A e. RR )
12		maxabs 11769	. . . . . 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 8207	. . . . . . . 8 |- ( A e. RR -> -u A e. CC )
15	14	addridd 8327	. . . . . . 7 |- ( A e. RR -> ( -u A + 0 ) = -u A )
16	14	subid1d 8478	. . . . . . . . 9 |- ( A e. RR -> ( -u A - 0 ) = -u A )
17	16	fveq2d 5643	. . . . . . . 8 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` -u A ) )
18	4	absnegd 11749	. . . . . . . 8 |- ( A e. RR -> ( abs ` -u A ) = ( abs ` A ) )
19	17, 18	eqtrd 2264	. . . . . . 7 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` A ) )
20	15, 19	oveq12d 6035	. . . . . 6 |- ( A e. RR -> ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) = ( -u A + ( abs ` A ) ) )
21	20	oveq1d 6032	. . . . 5 |- ( A e. RR -> ( ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) / 2 ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
22	13, 21	eqtrd 2264	. . . 4 |- ( A e. RR -> sup ( { -u A , 0 } , RR , < ) = ( ( -u A + ( abs ` A ) ) / 2 ) )
23	10, 22	oveq12d 6035	. . 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 11741	. . . . . 6 |- ( A e. RR -> ( abs ` A ) e. RR )
25	24	recnd 8207	. . . . 5 |- ( A e. RR -> ( abs ` A ) e. CC )
26	4, 25	addcld 8198	. . . 4 |- ( A e. RR -> ( A + ( abs ` A ) ) e. CC )
27	14, 25	addcld 8198	. . . 4 |- ( A e. RR -> ( -u A + ( abs ` A ) ) e. CC )
28		2cnd 9215	. . . 4 |- ( A e. RR -> 2 e. CC )
29		2ap0 9235	. . . . 5 |- 2 # 0
30	29	a1i 9	. . . 4 |- ( A e. RR -> 2 # 0 )
31	26, 27, 28, 30	divdirapd 9008	. . 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 8347	. . . . 5 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) )
33	4	negidd 8479	. . . . . 6 |- ( A e. RR -> ( A + -u A ) = 0 )
34	33	oveq1d 6032	. . . . 5 |- ( A e. RR -> ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) = ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) )
35	25, 25	addcld 8198	. . . . . 6 |- ( A e. RR -> ( ( abs ` A ) + ( abs ` A ) ) e. CC )
36	35	addlidd 8328	. . . . 5 |- ( A e. RR -> ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
37	32, 34, 36	3eqtrd 2268	. . . 4 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
38	37	oveq1d 6032	. . 3 |- ( A e. RR -> ( ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
39	23, 31, 38	3eqtr2d 2270	. 2 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
40	25	2timesd 9386	. . 3 |- ( A e. RR -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
41	40	oveq1d 6032	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
42	25, 28, 30	divcanap3d 8974	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( abs ` A ) )
43	39, 41, 42	3eqtr2d 2270	1 |- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( abs ` A ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {cpr 3670   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   supcsup 7180   RRcr 8030   0cc0 8031   + caddc 8034   x. cmul 8036   < clt 8213   - cmin 8349   -ucneg 8350   # cap 8760   / cdiv 8851   2c2 9193   abscabs 11557

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559

This theorem is referenced by: (None)