Theorem repiecele0 17049

Description: Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17051 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 3787	. . . . . . 7 |- ( x = A -> { x , 0 } = { A , 0 } )
3	2	infeq1d 7346	. . . . . 6 |- ( x = A -> inf ( { x , 0 } , RR , < ) = inf ( { A , 0 } , RR , < ) )
4	3	fveq2d 5697	. . . . 5 |- ( x = A -> ( F ` inf ( { x , 0 } , RR , < ) ) = ( F ` inf ( { A , 0 } , RR , < ) ) )
5	2	supeq1d 7321	. . . . . 6 |- ( x = A -> sup ( { x , 0 } , RR , < ) = sup ( { A , 0 } , RR , < ) )
6	5	fveq2d 5697	. . . . 5 |- ( x = A -> ( G ` sup ( { x , 0 } , RR , < ) ) = ( G ` sup ( { A , 0 } , RR , < ) ) )
7	4, 6	oveq12d 6097	. . . 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 6094	. . 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 /\ 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 17048	. . . 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 /\ A <_ 0 ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
15	1, 8, 9, 14	fvmptd3 5796	. 2 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
16		simp3 1030	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> A <_ 0 )
17		0re 8320	. . . . . . 7 |- 0 e. RR
18		mingeb 11991	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A <_ 0 <-> inf ( { A , 0 } , RR , < ) = A ) )
19	9, 17, 18	sylancl 417	. . . . . 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 5697	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` inf ( { A , 0 } , RR , < ) ) = ( F ` A ) )
22		maxleb 11965	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> ( A <_ 0 <-> sup ( { A , 0 } , RR , < ) = 0 ) )
23	9, 17, 22	sylancl 417	. . . . . . 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 5697	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( G ` 0 ) )
26	12	3ad2ant1 1049	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) = ( G ` 0 ) )
27	25, 26	eqtr4d 2274	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( F ` 0 ) )
28	21, 27	oveq12d 6097	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) = ( ( F ` A ) + ( F ` 0 ) ) )
29	28	oveq1d 6094	. 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 1049	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> F : ( -oo (,] 0 ) --> RR )
31		mnflt 10168	. . . . . . 7 |- ( A e. RR -> -oo < A )
32	9, 31	syl 14	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> -oo < A )
33		mnfxr 8376	. . . . . . 7 |- -oo e. RR*
34		elioc2 10321	. . . . . . 7 |- ( ( -oo e. RR* /\ 0 e. RR ) -> ( A e. ( -oo (,] 0 ) <-> ( A e. RR /\ -oo < A /\ A <_ 0 ) ) )
35	33, 17, 34	mp2an 430	. . . . . 6 |- ( A e. ( -oo (,] 0 ) <-> ( A e. RR /\ -oo < A /\ A <_ 0 ) )
36	9, 32, 16, 35	syl3anbrc 1212	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> A e. ( -oo (,] 0 ) )
37	30, 36	ffvelcdmd 5838	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` A ) e. RR )
38	37	recnd 8348	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` A ) e. CC )
39	11	3ad2ant1 1049	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> G : ( 0 [,) +oo ) --> RR )
40		maxcl 11959	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) e. RR )
41	9, 17, 40	sylancl 417	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) e. RR )
42		maxle2 11961	. . . . . . . 8 |- ( ( A e. RR /\ 0 e. RR ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
43	9, 17, 42	sylancl 417	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
44	41	ltpnfd 10166	. . . . . . 7 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) < +oo )
45		pnfxr 8372	. . . . . . . 8 |- +oo e. RR*
46		elico2 10322	. . . . . . . 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 430	. . . . . . 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 1212	. . . . . 6 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) )
49	39, 48	ffvelcdmd 5838	. . . . 5 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( G ` sup ( { A , 0 } , RR , < ) ) e. RR )
50	27, 49	eqeltrrd 2316	. . . 4 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) e. RR )
51	50	recnd 8348	. . 3 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( F ` 0 ) e. CC )
52	38, 51	pncand 8632	. 2 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( ( ( F ` A ) + ( F ` 0 ) ) - ( F ` 0 ) ) = ( F ` A ) )
53	15, 29, 52	3eqtrd 2275	1 |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( F ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cpr 3709   class class class wbr 4128   |-> cmpt 4190   -->wf 5371   ` cfv 5375  (class class class)co 6079   supcsup 7316  infcinf 7317   RRcr 8172   0cc0 8173   + caddc 8176   +oocpnf 8351   -oocmnf 8352   RR*cxr 8353   < clt 8354   <_ cle 8355   - cmin 8491   (,]cioc 10274   [,)cico 10275

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-rp 10038  df-ioc 10278  df-ico 10279  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748

This theorem is referenced by: (None)