Theorem max0addsup 11842

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 8222	. . . . . 6 |- 0 e. RR
2		maxabs 11832	. . . . . 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 8208	. . . . . . . 8 |- ( A e. RR -> A e. CC )
5	4	addridd 8370	. . . . . . 7 |- ( A e. RR -> ( A + 0 ) = A )
6	4	subid1d 8521	. . . . . . . 8 |- ( A e. RR -> ( A - 0 ) = A )
7	6	fveq2d 5652	. . . . . . 7 |- ( A e. RR -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
8	5, 7	oveq12d 6046	. . . . . 6 |- ( A e. RR -> ( ( A + 0 ) + ( abs ` ( A - 0 ) ) ) = ( A + ( abs ` A ) ) )
9	8	oveq1d 6043	. . . . 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 8482	. . . . . 6 |- ( A e. RR -> -u A e. RR )
12		maxabs 11832	. . . . . 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 8250	. . . . . . . 8 |- ( A e. RR -> -u A e. CC )
15	14	addridd 8370	. . . . . . 7 |- ( A e. RR -> ( -u A + 0 ) = -u A )
16	14	subid1d 8521	. . . . . . . . 9 |- ( A e. RR -> ( -u A - 0 ) = -u A )
17	16	fveq2d 5652	. . . . . . . 8 |- ( A e. RR -> ( abs ` ( -u A - 0 ) ) = ( abs ` -u A ) )
18	4	absnegd 11812	. . . . . . . 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 6046	. . . . . 6 |- ( A e. RR -> ( ( -u A + 0 ) + ( abs ` ( -u A - 0 ) ) ) = ( -u A + ( abs ` A ) ) )
21	20	oveq1d 6043	. . . . 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 6046	. . 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 11804	. . . . . 6 |- ( A e. RR -> ( abs ` A ) e. RR )
25	24	recnd 8250	. . . . 5 |- ( A e. RR -> ( abs ` A ) e. CC )
26	4, 25	addcld 8241	. . . 4 |- ( A e. RR -> ( A + ( abs ` A ) ) e. CC )
27	14, 25	addcld 8241	. . . 4 |- ( A e. RR -> ( -u A + ( abs ` A ) ) e. CC )
28		2cnd 9258	. . . 4 |- ( A e. RR -> 2 e. CC )
29		2ap0 9278	. . . . 5 |- 2 # 0
30	29	a1i 9	. . . 4 |- ( A e. RR -> 2 # 0 )
31	26, 27, 28, 30	divdirapd 9051	. . 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 8390	. . . . 5 |- ( A e. RR -> ( ( A + ( abs ` A ) ) + ( -u A + ( abs ` A ) ) ) = ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) )
33	4	negidd 8522	. . . . . 6 |- ( A e. RR -> ( A + -u A ) = 0 )
34	33	oveq1d 6043	. . . . 5 |- ( A e. RR -> ( ( A + -u A ) + ( ( abs ` A ) + ( abs ` A ) ) ) = ( 0 + ( ( abs ` A ) + ( abs ` A ) ) ) )
35	25, 25	addcld 8241	. . . . . 6 |- ( A e. RR -> ( ( abs ` A ) + ( abs ` A ) ) e. CC )
36	35	addlidd 8371	. . . . 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 6043	. . 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 9429	. . 3 |- ( A e. RR -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
41	40	oveq1d 6043	. 2 |- ( A e. RR -> ( ( 2 x. ( abs ` A ) ) / 2 ) = ( ( ( abs ` A ) + ( abs ` A ) ) / 2 ) )
42	25, 28, 30	divcanap3d 9017	. 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 1398   e. wcel 2202   {cpr 3674   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   supcsup 7224   RRcr 8074   0cc0 8075   + caddc 8078   x. cmul 8080   < clt 8256   - cmin 8392   -ucneg 8393   # cap 8803   / cdiv 8894   2c2 9236   abscabs 11620

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622

This theorem is referenced by: (None)