Theorem repiecege0 16860

Description: Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 16861 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 3770	. . . . . . 7 |- ( x = A -> { x , 0 } = { A , 0 } )
3	2	infeq1d 7305	. . . . . 6 |- ( x = A -> inf ( { x , 0 } , RR , < ) = inf ( { A , 0 } , RR , < ) )
4	3	fveq2d 5676	. . . . 5 |- ( x = A -> ( F ` inf ( { x , 0 } , RR , < ) ) = ( F ` inf ( { A , 0 } , RR , < ) ) )
5	2	supeq1d 7280	. . . . . 6 |- ( x = A -> sup ( { x , 0 } , RR , < ) = sup ( { A , 0 } , RR , < ) )
6	5	fveq2d 5676	. . . . 5 |- ( x = A -> ( G ` sup ( { x , 0 } , RR , < ) ) = ( G ` sup ( { A , 0 } , RR , < ) ) )
7	4, 6	oveq12d 6070	. . . 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 6067	. . 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 /\ 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 16858	. . . 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 /\ 0 <_ A ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
15	1, 8, 9, 14	fvmptd3 5773	. 2 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
16		mincom 11922	. . . . . 6 |- inf ( { 0 , A } , RR , < ) = inf ( { A , 0 } , RR , < )
17		simp3 1026	. . . . . . 7 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> 0 <_ A )
18		0re 8279	. . . . . . . 8 |- 0 e. RR
19		mingeb 11935	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> inf ( { 0 , A } , RR , < ) = 0 ) )
20	18, 9, 19	sylancr 414	. . . . . . 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 2281	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> inf ( { A , 0 } , RR , < ) = 0 )
23	22	fveq2d 5676	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` inf ( { A , 0 } , RR , < ) ) = ( F ` 0 ) )
24		maxcom 11896	. . . . . 6 |- sup ( { 0 , A } , RR , < ) = sup ( { A , 0 } , RR , < )
25		maxleb 11909	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> sup ( { 0 , A } , RR , < ) = A ) )
26	18, 9, 25	sylancr 414	. . . . . . 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 2281	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> sup ( { A , 0 } , RR , < ) = A )
29	28	fveq2d 5676	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` sup ( { A , 0 } , RR , < ) ) = ( G ` A ) )
30	23, 29	oveq12d 6070	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) = ( ( F ` 0 ) + ( G ` A ) ) )
31	30	oveq1d 6067	. 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 1045	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> F : ( -oo (,] 0 ) --> RR )
33		mnfxr 8335	. . . . . . 7 |- -oo e. RR*
34		0xr 8325	. . . . . . 7 |- 0 e. RR*
35		mnflt0 10123	. . . . . . 7 |- -oo < 0
36		ubioc1 10268	. . . . . . 7 |- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) )
37	33, 34, 35, 36	mp3an 1374	. . . . . 6 |- 0 e. ( -oo (,] 0 )
38	37	a1i 9	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> 0 e. ( -oo (,] 0 ) )
39	32, 38	ffvelcdmd 5815	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` 0 ) e. RR )
40	39	recnd 8307	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( F ` 0 ) e. CC )
41	11	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> G : ( 0 [,) +oo ) --> RR )
42	9	ltpnfd 10120	. . . . . 6 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> A < +oo )
43		pnfxr 8331	. . . . . . 7 |- +oo e. RR*
44		elico2 10276	. . . . . . 7 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A /\ A < +oo ) ) )
45	18, 43, 44	mp2an 426	. . . . . 6 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A /\ A < +oo ) )
46	9, 17, 42, 45	syl3anbrc 1208	. . . . 5 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> A e. ( 0 [,) +oo ) )
47	41, 46	ffvelcdmd 5815	. . . 4 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` A ) e. RR )
48	47	recnd 8307	. . 3 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( G ` A ) e. CC )
49	40, 48	pncan2d 8591	. 2 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( ( ( F ` 0 ) + ( G ` A ) ) - ( F ` 0 ) ) = ( G ` A ) )
50	15, 31, 49	3eqtrd 2271	1 |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( G ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {cpr 3692   class class class wbr 4111   |-> cmpt 4173   -->wf 5350   ` cfv 5354  (class class class)co 6052   supcsup 7275  infcinf 7276   RRcr 8131   0cc0 8132   + caddc 8135   +oocpnf 8310   -oocmnf 8311   RR*cxr 8312   < clt 8313   <_ cle 8314   - cmin 8449   (,]cioc 10228   [,)cico 10229

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-rp 9993  df-ioc 10232  df-ico 10233  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692

This theorem is referenced by: (None)