Theorem dvidrelem 15374

Description: Lemma for dvidre 15379 and dvconstre 15378. Analogue of dvidlemap 15373 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 8141	. . . . . . 7 |- RR e. _V
3		cnex 8131	. . . . . . 7 |- CC e. _V
4	2, 3	fpm 6836	. . . . . 6 |- ( F : RR --> CC -> F e. ( CC ^pm RR ) )
5	1, 4	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm RR ) )
6		dvfpm 15371	. . . . 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 8099	. . . . . . . 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 15367	. . . . . 6 |- ( ph -> dom ( RR _D F ) C_ RR )
12		reldvg 15361	. . . . . . . . 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 15206	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
17		uniretop 15207	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
18	17	ntrtop 14810	. . . . . . . . . 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 15348	. . . . . . . . . 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 15278	. . . . . . . . . . . 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 15356	. . . . . . . . . 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 5053	. . . . . . . . . . . 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 6022	. . . . . . . . 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 15239	. . . . . . . . 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 15364	. . . . . . . 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 4959	. . . . . . 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 5461	. . . 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 5474	. 2 |- ( ph -> ( RR _D F ) Fn RR )
56		fnconstg 5525	. . 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 5477	. . . . 5 |- ( ( ph /\ x e. RR ) -> Fun ( RR _D F ) )
60		funbrfvb 5676	. . . . 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 5857	. . . 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 5737	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 4717   dom cdm 4719   ran crn 4720   |` cres 4721   o. ccom 4723   Rel wrel 4724   Fun wfun 5312   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6007   ^pm cpm 6804   CCcc 8005   RRcr 8006   - cmin 8325   # cap 8736   / cdiv 8827   (,)cioo 10092   abscabs 11516   topGenctg 13295   MetOpencmopn 14513   Topctop 14679   intcnt 14775   -cn->ccncf 15252   lim CC climc 15336   _D cdv 15337

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

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 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-map 6805  df-pm 6806  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-xneg 9976  df-xadd 9977  df-ioo 10096  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-rest 13282  df-topgen 13301  df-psmet 14515  df-xmet 14516  df-met 14517  df-bl 14518  df-mopn 14519  df-top 14680  df-topon 14693  df-bases 14725  df-ntr 14778  df-cn 14870  df-cnp 14871  df-cncf 15253  df-limced 15338  df-dvap 15339

This theorem is referenced by:  dvconstre  15378  dvidre  15379