Theorem dvidrelem 15387

Description: Lemma for dvidre 15392 and dvconstre 15391. Analogue of dvidlemap 15386 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)

Hypotheses

Ref	Expression
dvidrelem.1	|- ( ph -> F : RR --> CC )
dvidrelem.2	|- ( ( ph /\ ( x e. RR /\ z e. RR /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
dvidrelem.3	|- B e. CC

Assertion

Ref	Expression
dvidrelem	|- ( ph -> ( RR _D F ) = ( RR X. { B } ) )

Distinct variable groups:   x, B, z   x, F, z   ph, x, z

Proof of Theorem dvidrelem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvidrelem.1	. . . . . 6 |- ( ph -> F : RR --> CC )
2		reex 8149	. . . . . . 7 |- RR e. _V
3		cnex 8139	. . . . . . 7 |- CC e. _V
4	2, 3	fpm 6841	. . . . . 6 |- ( F : RR --> CC -> F e. ( CC ^pm RR ) )
5	1, 4	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm RR ) )
6		dvfpm 15384	. . . . 5 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
7	5, 6	syl 14	. . . 4 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> CC )
8		ax-resscn 8107	. . . . . . . 8 |- RR C_ CC
9	8	a1i 9	. . . . . . 7 |- ( ph -> RR C_ CC )
10		ssidd 3245	. . . . . . 7 |- ( ph -> RR C_ RR )
11	9, 1, 10	dvbss 15380	. . . . . 6 |- ( ph -> dom ( RR _D F ) C_ RR )
12		reldvg 15374	. . . . . . . . 9 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> Rel ( RR _D F ) )
13	9, 5, 12	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( RR _D F ) )
14	13	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> Rel ( RR _D F ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> x e. RR )
16		retop 15219	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
17		uniretop 15220	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
18	17	ntrtop 14823	. . . . . . . . . 10 |- ( ( topGen ` ran (,) ) e. Top -> ( ( int ` ( topGen ` ran (,) ) ) ` RR ) = RR )
19	16, 18	ax-mp 5	. . . . . . . . 9 |- ( ( int ` ( topGen ` ran (,) ) ) ` RR ) = RR
20	15, 19	eleqtrrdi 2323	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` RR ) )
21		limcresi 15361	. . . . . . . . . 10 |- ( ( z e. RR |-> B ) lim CC x ) C_ ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x )
22		dvidrelem.3	. . . . . . . . . . . 12 |- B e. CC
23		ssidd 3245	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> CC C_ CC )
24		cncfmptc 15291	. . . . . . . . . . . 12 |- ( ( B e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
25	22, 8, 23, 24	mp3an12i 1375	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
26		eqidd 2230	. . . . . . . . . . 11 |- ( z = x -> B = B )
27	25, 15, 26	cnmptlimc 15369	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> B e. ( ( z e. RR |-> B ) lim CC x ) )
28	21, 27	sselid 3222	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> B e. ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x ) )
29		breq1 4086	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
30	29	elrab 2959	. . . . . . . . . . . . 13 |- ( z e. { w e. RR | w # x } <-> ( z e. RR /\ z # x ) )
31		dvidrelem.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. RR /\ z e. RR /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
32	31	3exp2 1249	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. RR -> ( z e. RR -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
33	32	imp43 355	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR ) /\ ( z e. RR /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
34	30, 33	sylan2b 287	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR ) /\ z e. { w e. RR | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
35	34	mpteq2dva 4174	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. RR | w # x } |-> B ) )
36		ssrab2 3309	. . . . . . . . . . . 12 |- { w e. RR | w # x } C_ RR
37		resmpt 5056	. . . . . . . . . . . 12 |- ( { w e. RR | w # x } C_ RR -> ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) = ( z e. { w e. RR | w # x } |-> B ) )
38	36, 37	ax-mp 5	. . . . . . . . . . 11 |- ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) = ( z e. { w e. RR | w # x } |-> B )
39	35, 38	eqtr4di 2280	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) )
40	39	oveq1d 6025	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x ) )
41	28, 40	eleqtrrd 2309	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> B e. ( ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
42		eqid 2229	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
43	42	tgioo2cntop 15252	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
44		eqid 2229	. . . . . . . . 9 |- ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
45	8	a1i 9	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> RR C_ CC )
46	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> F : RR --> CC )
47		ssidd 3245	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> RR C_ RR )
48	43, 42, 44, 45, 46, 47	eldvap 15377	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( x ( RR _D F ) B <-> ( x e. ( ( int ` ( topGen ` ran (,) ) ) ` RR ) /\ B e. ( ( z e. { w e. RR | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
49	20, 41, 48	mpbir2and 950	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> x ( RR _D F ) B )
50		releldm 4962	. . . . . . 7 |- ( ( Rel ( RR _D F ) /\ x ( RR _D F ) B ) -> x e. dom ( RR _D F ) )
51	14, 49, 50	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. RR ) -> x e. dom ( RR _D F ) )
52	11, 51	eqelssd 3243	. . . . 5 |- ( ph -> dom ( RR _D F ) = RR )
53	52	feq2d 5464	. . . 4 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : RR --> CC ) )
54	7, 53	mpbid 147	. . 3 |- ( ph -> ( RR _D F ) : RR --> CC )
55	54	ffnd 5477	. 2 |- ( ph -> ( RR _D F ) Fn RR )
56		fnconstg 5528	. . 3 |- ( B e. CC -> ( RR X. { B } ) Fn RR )
57	22, 56	mp1i 10	. 2 |- ( ph -> ( RR X. { B } ) Fn RR )
58	7	adantr 276	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
59	58	ffund 5480	. . . . 5 |- ( ( ph /\ x e. RR ) -> Fun ( RR _D F ) )
60		funbrfvb 5679	. . . . 5 |- ( ( Fun ( RR _D F ) /\ x e. dom ( RR _D F ) ) -> ( ( ( RR _D F ) ` x ) = B <-> x ( RR _D F ) B ) )
61	59, 51, 60	syl2anc 411	. . . 4 |- ( ( ph /\ x e. RR ) -> ( ( ( RR _D F ) ` x ) = B <-> x ( RR _D F ) B ) )
62	49, 61	mpbird 167	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) = B )
63	22	a1i 9	. . . 4 |- ( ph -> B e. CC )
64		fvconst2g 5860	. . . 4 |- ( ( B e. CC /\ x e. RR ) -> ( ( RR X. { B } ) ` x ) = B )
65	63, 64	sylan 283	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( RR X. { B } ) ` x ) = B )
66	62, 65	eqtr4d 2265	. 2 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) = ( ( RR X. { B } ) ` x ) )
67	55, 57, 66	eqfnfvd 5740	1 |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   {csn 3666   class class class wbr 4083   |-> cmpt 4145   X. cxp 4718   dom cdm 4720   ran crn 4721   |` cres 4722   o. ccom 4724   Rel wrel 4725   Fun wfun 5315   Fn wfn 5316   -->wf 5317   ` cfv 5321  (class class class)co 6010   ^pm cpm 6809   CCcc 8013   RRcr 8014   - cmin 8333   # cap 8744   / cdiv 8835   (,)cioo 10101   abscabs 11529   topGenctg 13308   MetOpencmopn 14526   Topctop 14692   intcnt 14788   -cn->ccncf 15265   lim CC climc 15349   _D cdv 15350

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-map 6810  df-pm 6811  df-sup 7167  df-inf 7168  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-xneg 9985  df-xadd 9986  df-ioo 10105  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-rest 13295  df-topgen 13314  df-psmet 14528  df-xmet 14529  df-met 14530  df-bl 14531  df-mopn 14532  df-top 14693  df-topon 14706  df-bases 14738  df-ntr 14791  df-cn 14883  df-cnp 14884  df-cncf 15266  df-limced 15351  df-dvap 15352

This theorem is referenced by:  dvconstre  15391  dvidre  15392