Theorem repiecege0 17294

Description: Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17295 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)

Hypotheses

Ref	Expression
repiece.f	|- ( ph -> F : ( -oo (,] 0 ) --> RR )
repiece.g	|- ( ph -> G : ( 0 [,) +oo ) --> RR )
repiece.0	|- ( ph -> ( F ` 0 ) = ( G ` 0 ) )
repiece.h	|- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )

Assertion

Ref	Expression
repiecege0	|- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( G ` A ) )

Distinct variable groups:   x, A   x, F   x, G

Allowed substitution hints:   ph( x)   H( x)

Proof of Theorem repiecege0

Step	Hyp	Ref	Expression
1		repiece.h	. . 3 |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
2		preq1 3788	. . . . . . 7 |- ( x = A -> { x , 0 } = { A , 0 } )
3	2	infeq1d 7359	. . . . . 6 |- ( x = A -> inf ( { x , 0 } , RR , < ) = inf ( { A , 0 } , RR , < ) )
4	3	fveq2d 5702	. . . . 5 |- ( x = A -> ( F ` inf ( { x , 0 } , RR , < ) ) = ( F ` inf ( { A , 0 } , RR , < ) ) )
5	2	supeq1d 7334	. . . . . 6 |- ( x = A -> sup ( { x , 0 } , RR , < ) = sup ( { A , 0 } , RR , < ) )
6	5	fveq2d 5702	. . . . 5 |- ( x = A -> ( G ` sup ( { x , 0 } , RR , < ) ) = ( G ` sup ( { A , 0 } , RR , < ) ) )
7	4, 6	oveq12d 6106	. . . 4 |- ( x = A -> ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) = ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) )
8	7	oveq1d 6103	. . 3 |- ( x = A -> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) = ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
9		simp2 1029	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> A e. RR )
10		repiece.f	. . . . 5 |- ( ph -> F : ( -oo (,] 0 ) --> RR )
11		repiece.g	. . . . 5 |- ( ph -> G : ( 0 [,) +oo ) --> RR )
12		repiece.0	. . . . 5 |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )
13	10, 11, 12, 1	repiecelem 17292	. . . 4 |- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
14	13	3adant3 1048	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
15	1, 8, 9, 14	fvmptd3 5802	. 2 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
16		mincom 12020	. . . . . 6 |- inf ( { 0 , A } , RR , < ) = inf ( { A , 0 } , RR , < )
17		simp3 1030	. . . . . . 7 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> 0 <_ A )
18		0re 8333	. . . . . . . 8 |- 0 e. RR
19		mingeb 12034	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> inf ( { 0 , A } , RR , < ) = 0 ) )
20	18, 9, 19	sylancr 418	. . . . . . 7 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( 0 <_ A <-> inf ( { 0 , A } , RR , < ) = 0 ) )
21	17, 20	mpbid 147	. . . . . 6 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> inf ( { 0 , A } , RR , < ) = 0 )
22	16, 21	eqtr3id 2285	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> inf ( { A , 0 } , RR , < ) = 0 )
23	22	fveq2d 5702	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` inf ( { A , 0 } , RR , < ) ) = ( F ` 0 ) )
24		maxcom 11993	. . . . . 6 |- sup ( { 0 , A } , RR , < ) = sup ( { A , 0 } , RR , < )
25		maxleb 12006	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> sup ( { 0 , A } , RR , < ) = A ) )
26	18, 9, 25	sylancr 418	. . . . . . 7 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( 0 <_ A <-> sup ( { 0 , A } , RR , < ) = A ) )
27	17, 26	mpbid 147	. . . . . 6 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> sup ( { 0 , A } , RR , < ) = A )
28	24, 27	eqtr3id 2285	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> sup ( { A , 0 } , RR , < ) = A )
29	28	fveq2d 5702	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( G ` A ) )
30	23, 29	oveq12d 6106	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) = ( ( F ` 0 ) + ( G ` A ) ) )
31	30	oveq1d 6103	. 2 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) = ( ( ( F ` 0 ) + ( G ` A ) ) - ( F ` 0 ) ) )
32	10	3ad2ant1 1049	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> F : ( -oo (,] 0 ) --> RR )
33		mnfxr 8389	. . . . . . 7 |- -oo e. RR*
34		0xr 8379	. . . . . . 7 |- 0 e. RR*
35		mnflt0 10204	. . . . . . 7 |- -oo < 0
36		ubioc1 10349	. . . . . . 7 |- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) )
37	33, 34, 35, 36	mp3an 1378	. . . . . 6 |- 0 e. ( -oo (,] 0 )
38	37	a1i 9	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> 0 e. ( -oo (,] 0 ) )
39	32, 38	ffvelcdmd 5847	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` 0 ) e. RR )
40	39	recnd 8361	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` 0 ) e. CC )
41	11	3ad2ant1 1049	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> G : ( 0 [,) +oo ) --> RR )
42	9	ltpnfd 10201	. . . . . 6 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> A < +oo )
43		pnfxr 8385	. . . . . . 7 |- +oo e. RR*
44		elico2 10357	. . . . . . 7 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A /\ A < +oo ) ) )
45	18, 43, 44	mp2an 430	. . . . . 6 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A /\ A < +oo ) )
46	9, 17, 42, 45	syl3anbrc 1212	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> A e. ( 0 [,) +oo ) )
47	41, 46	ffvelcdmd 5847	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` A ) e. RR )
48	47	recnd 8361	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` A ) e. CC )
49	40, 48	pncan2d 8647	. 2 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( ( F ` 0 ) + ( G ` A ) ) - ( F ` 0 ) ) = ( G ` A ) )
50	15, 31, 49	3eqtrd 2275	1 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( G ` A ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cpr 3710   class class class wbr 4131   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   supcsup 7329  infcinf 7330   RRcr 8185   0cc0 8186   + caddc 8189   +oocpnf 8364   -oocmnf 8365   RR*cxr 8366   < clt 8367   <_ cle 8368   - cmin 8505   (,]cioc 10309   [,)cico 10310

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-rp 10073  df-ioc 10313  df-ico 10314  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788

This theorem is used by: (None)