Theorem repiecele0 16949

Description: Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 16951 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
repiecele0	|- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( F ` A ) )

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

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

Proof of Theorem repiecele0

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 3773	. . . . . . 7 |- ( x = A -> { x , 0 } = { A , 0 } )
3	2	infeq1d 7316	. . . . . 6 |- ( x = A -> inf ( { x , 0 } , RR , < ) = inf ( { A , 0 } , RR , < ) )
4	3	fveq2d 5679	. . . . 5 |- ( x = A -> ( F ` inf ( { x , 0 } , RR , < ) ) = ( F ` inf ( { A , 0 } , RR , < ) ) )
5	2	supeq1d 7291	. . . . . 6 |- ( x = A -> sup ( { x , 0 } , RR , < ) = sup ( { A , 0 } , RR , < ) )
6	5	fveq2d 5679	. . . . 5 |- ( x = A -> ( G ` sup ( { x , 0 } , RR , < ) ) = ( G ` sup ( { A , 0 } , RR , < ) ) )
7	4, 6	oveq12d 6076	. . . 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 6073	. . 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 1025	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> 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 16948	. . . 4 |- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
14	13	3adant3 1044	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
15	1, 8, 9, 14	fvmptd3 5776	. 2 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
16		simp3 1026	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> A <_ 0 )
17		0re 8290	. . . . . . 7 |- 0 e. RR
18		mingeb 11955	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A <_ 0 <-> inf ( { A , 0 } , RR , < ) = A ) )
19	9, 17, 18	sylancl 413	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( A <_ 0 <-> inf ( { A , 0 } , RR , < ) = A ) )
20	16, 19	mpbid 147	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> inf ( { A , 0 } , RR , < ) = A )
21	20	fveq2d 5679	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` inf ( { A , 0 } , RR , < ) ) = ( F ` A ) )
22		maxleb 11929	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> ( A <_ 0 <-> sup ( { A , 0 } , RR , < ) = 0 ) )
23	9, 17, 22	sylancl 413	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( A <_ 0 <-> sup ( { A , 0 } , RR , < ) = 0 ) )
24	16, 23	mpbid 147	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) = 0 )
25	24	fveq2d 5679	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( G ` 0 ) )
26	12	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) = ( G ` 0 ) )
27	25, 26	eqtr4d 2270	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( F ` 0 ) )
28	21, 27	oveq12d 6076	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) = ( ( F ` A ) + ( F ` 0 ) ) )
29	28	oveq1d 6073	. 2 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) = ( ( ( F ` A ) + ( F ` 0 ) ) - ( F ` 0 ) ) )
30	10	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> F : ( -oo (,] 0 ) --> RR )
31		mnflt 10138	. . . . . . 7 |- ( A e. RR -> -oo < A )
32	9, 31	syl 14	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> -oo < A )
33		mnfxr 8346	. . . . . . 7 |- -oo e. RR*
34		elioc2 10291	. . . . . . 7 |- ( ( -oo e. RR* /\ 0 e. RR ) -> ( A e. ( -oo (,] 0 ) <-> ( A e. RR /\ -oo < A /\ A <_ 0 ) ) )
35	33, 17, 34	mp2an 426	. . . . . 6 |- ( A e. ( -oo (,] 0 ) <-> ( A e. RR /\ -oo < A /\ A <_ 0 ) )
36	9, 32, 16, 35	syl3anbrc 1208	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> A e. ( -oo (,] 0 ) )
37	30, 36	ffvelcdmd 5818	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` A ) e. RR )
38	37	recnd 8318	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` A ) e. CC )
39	11	3ad2ant1 1045	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> G : ( 0 [,) +oo ) --> RR )
40		maxcl 11923	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) e. RR )
41	9, 17, 40	sylancl 413	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) e. RR )
42		maxle2 11925	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
43	9, 17, 42	sylancl 413	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
44	41	ltpnfd 10136	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) < +oo )
45		pnfxr 8342	. . . . . . . 8 |- +oo e. RR*
46		elico2 10292	. . . . . . . 8 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) <-> ( sup ( { A , 0 } , RR , < ) e. RR /\ 0 <_ sup ( { A , 0 } , RR , < ) /\ sup ( { A , 0 } , RR , < ) < +oo ) ) )
47	17, 45, 46	mp2an 426	. . . . . . 7 |- ( sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) <-> ( sup ( { A , 0 } , RR , < ) e. RR /\ 0 <_ sup ( { A , 0 } , RR , < ) /\ sup ( { A , 0 } , RR , < ) < +oo ) )
48	41, 43, 44, 47	syl3anbrc 1208	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) )
49	39, 48	ffvelcdmd 5818	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) e. RR )
50	27, 49	eqeltrrd 2312	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) e. RR )
51	50	recnd 8318	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) e. CC )
52	38, 51	pncand 8602	. 2 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( ( F ` A ) + ( F ` 0 ) ) - ( F ` 0 ) ) = ( F ` A ) )
53	15, 29, 52	3eqtrd 2271	1 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( F ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {cpr 3695   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   supcsup 7286  infcinf 7287   RRcr 8142   0cc0 8143   + caddc 8146   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   <_ cle 8325   - cmin 8461   (,]cioc 10244   [,)cico 10245

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-isom 5366  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-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-z 9598  df-uz 9875  df-rp 10008  df-ioc 10248  df-ico 10249  df-seqfrec 10837  df-exp 10928  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712

This theorem is referenced by: (None)