Theorem repiecelem 16796

Description: Lemma for repiecele0 16797, repiecege0 16798, and repiecef 16799. 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 8270	. . . . . 6 |- 0 e. RR
5		mincl 11909	. . . . . 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 10112	. . . . . 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 11911	. . . . . 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 8326	. . . . . 6 |- -oo e. RR*
12		elioc2 10265	. . . . . 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 1208	. . . 4 |- ( ( ph /\ A e. RR ) -> inf ( { A , 0 } , RR , < ) e. ( -oo (,] 0 ) )
15	2, 14	ffvelcdmd 5812	. . 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 11888	. . . . . 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 11890	. . . . . 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 10110	. . . . 5 |- ( ( ph /\ A e. RR ) -> sup ( { A , 0 } , RR , < ) < +oo )
23		pnfxr 8322	. . . . . 6 |- +oo e. RR*
24		elico2 10266	. . . . . 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 1208	. . . 4 |- ( ( ph /\ A e. RR ) -> sup ( { A , 0 } , RR , < ) e. ( 0 [,) +oo ) )
27	17, 26	ffvelcdmd 5812	. . 3 |- ( ( ph /\ A e. RR ) -> ( G ` sup ( { A , 0 } , RR , < ) ) e. RR )
28	15, 27	readdcld 8299	. 2 |- ( ( ph /\ A e. RR ) -> ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) e. RR )
29		0xr 8316	. . . . 5 |- 0 e. RR*
30		mnflt0 10113	. . . . 5 |- -oo < 0
31		ubioc1 10258	. . . . 5 |- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) )
32	11, 29, 30, 31	mp3an 1374	. . . 4 |- 0 e. ( -oo (,] 0 )
33	32	a1i 9	. . 3 |- ( ( ph /\ A e. RR ) -> 0 e. ( -oo (,] 0 ) )
34	2, 33	ffvelcdmd 5812	. 2 |- ( ( ph /\ A e. RR ) -> ( F ` 0 ) e. RR )
35	28, 34	resubcld 8650	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 1005   = wceq 1398   e. wcel 2203   {cpr 3689   class class class wbr 4108   |-> cmpt 4170   -->wf 5347   ` cfv 5351  (class class class)co 6049   supcsup 7272  infcinf 7273   RRcr 8122   0cc0 8123   + caddc 8126   +oocpnf 8301   -oocmnf 8302   RR*cxr 8303   < clt 8304   <_ cle 8305   - cmin 8440   (,]cioc 10218   [,)cico 10219

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-inf 7275  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-z 9574  df-uz 9850  df-rp 9983  df-ioc 10222  df-ico 10223  df-seqfrec 10806  df-exp 10897  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677

This theorem is referenced by:  repiecele0  16797  repiecege0  16798  repiecef  16799