Theorem repiecelem 16696

Description: Lemma for repiecele0 16697, repiecege0 16698, and repiecef 16699. The function H is defined everywhere. (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
repiecelem	|- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )

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

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

Proof of Theorem repiecelem

Step	Hyp	Ref	Expression
1		repiece.f	. . . . 5 |- ( ph -> F : ( -oo (,] 0 ) --> RR )
2	1	adantr 276	. . . 4 |- ( ( ph /\ A e. RR ) -> F : ( -oo (,] 0 ) --> RR )
3		simpr 110	. . . . . 6 |- ( ( ph /\ A e. RR ) -> A e. RR )
4		0re 8184	. . . . . 6 |- 0 e. RR
5		mincl 11814	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> inf ( { A , 0 } , RR , < ) e. RR )
6	3, 4, 5	sylancl 413	. . . . 5 |- ( ( ph /\ A e. RR ) -> inf ( { A , 0 } , RR , < ) e. RR )
7		mnflt 10023	. . . . . 6 |- (inf ( { A , 0 } , RR , < ) e. RR -> -oo < inf ( { A , 0 } , RR , < ) )
8	6, 7	syl 14	. . . . 5 |- ( ( ph /\ A e. RR ) -> -oo < inf ( { A , 0 } , RR , < ) )
9		min2inf 11816	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> inf ( { A , 0 } , RR , < ) <_ 0 )
10	3, 4, 9	sylancl 413	. . . . 5 |- ( ( ph /\ A e. RR ) -> inf ( { A , 0 } , RR , < ) <_ 0 )
11		mnfxr 8241	. . . . . 6 |- -oo e. RR*
12		elioc2 10176	. . . . . 6 |- ( ( -oo e. RR* /\ 0 e. RR ) -> (inf ( { A , 0 } , RR , < ) e. ( -oo (,] 0 ) <-> (inf ( { A , 0 } , RR , < ) e. RR /\ -oo < inf ( { A , 0 } , RR , < ) /\ inf ( { A , 0 } , RR , < ) <_ 0 ) ) )
13	11, 4, 12	mp2an 426	. . . . 5 |- (inf ( { A , 0 } , RR , < ) e. ( -oo (,] 0 ) <-> (inf ( { A , 0 } , RR , < ) e. RR /\ -oo < inf ( { A , 0 } , RR , < ) /\ inf ( { A , 0 } , RR , < ) <_ 0 ) )
14	6, 8, 10, 13	syl3anbrc 1207	. . . 4 |- ( ( ph /\ A e. RR ) -> inf ( { A , 0 } , RR , < ) e. ( -oo (,] 0 ) )
15	2, 14	ffvelcdmd 5786	. . 3 |- ( ( ph /\ A e. RR ) -> ( F ` inf ( { A , 0 } , RR , < ) ) e. RR )
16		repiece.g	. . . . 5 |- ( ph -> G : ( 0 [,) +oo ) --> RR )
17	16	adantr 276	. . . 4 |- ( ( ph /\ A e. RR ) -> G : ( 0 [,) +oo ) --> RR )
18		maxcl 11793	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> sup ( { A , 0 } , RR , < ) e. RR )
19	3, 4, 18	sylancl 413	. . . . 5 |- ( ( ph /\ A e. RR ) -> sup ( { A , 0 } , RR , < ) e. RR )
20		maxle2 11795	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
21	3, 4, 20	sylancl 413	. . . . 5 |- ( ( ph /\ A e. RR ) -> 0 <_ sup ( { A , 0 } , RR , < ) )
22	19	ltpnfd 10021	. . . . 5 |- ( ( ph /\ A e. RR ) -> sup ( { A , 0 } , RR , < ) < +oo )
23		pnfxr 8237	. . . . . 6 |- +oo e. RR*
24		elico2 10177	. . . . . 6 |- ( ( 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 ) ) )
25	4, 23, 24	mp2an 426	. . . . 5 |- ( sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) <-> ( sup ( { A , 0 } , RR , < ) e. RR /\ 0 <_ sup ( { A , 0 } , RR , < ) /\ sup ( { A , 0 } , RR , < ) < +oo ) )
26	19, 21, 22, 25	syl3anbrc 1207	. . . 4 |- ( ( ph /\ A e. RR ) -> sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) )
27	17, 26	ffvelcdmd 5786	. . 3 |- ( ( ph /\ A e. RR ) -> ( G ` sup ( { A , 0 } , RR , < ) ) e. RR )
28	15, 27	readdcld 8214	. 2 |- ( ( ph /\ A e. RR ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) e. RR )
29		0xr 8231	. . . . 5 |- 0 e. RR*
30		mnflt0 10024	. . . . 5 |- -oo < 0
31		ubioc1 10169	. . . . 5 |- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) )
32	11, 29, 30, 31	mp3an 1373	. . . 4 |- 0 e. ( -oo (,] 0 )
33	32	a1i 9	. . 3 |- ( ( ph /\ A e. RR ) -> 0 e. ( -oo (,] 0 ) )
34	2, 33	ffvelcdmd 5786	. 2 |- ( ( ph /\ A e. RR ) -> ( F ` 0 ) e. RR )
35	28, 34	resubcld 8565	1 |- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   {cpr 3671   class class class wbr 4089   |-> cmpt 4151   -->wf 5324   ` cfv 5328  (class class class)co 6023   supcsup 7186  infcinf 7187   RRcr 8036   0cc0 8037   + caddc 8040   +oocpnf 8216   -oocmnf 8217   RR*cxr 8218   < clt 8219   <_ cle 8220   - cmin 8355   (,]cioc 10129   [,)cico 10130

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-sup 7188  df-inf 7189  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-rp 9894  df-ioc 10133  df-ico 10134  df-seqfrec 10716  df-exp 10807  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582

This theorem is referenced by:  repiecele0  16697  repiecege0  16698  repiecef  16699