Theorem dvidrelem 15776

Description: Lemma for dvidre 15781 and dvconstre 15780. Analogue of dvidlemap 15775 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 8307	. . . . . . 7 |- RR e. _V
3		cnex 8297	. . . . . . 7 |- CC e. _V
4	2, 3	fpm 6956	. . . . . 6 |- ( F : RR --> CC -> F e. ( CC ^pm RR ) )
5	1, 4	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm RR ) )
6		dvfpm 15773	. . . . 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 8265	. . . . . . . 8 |- RR C_ CC
9	8	a1i 9	. . . . . . 7 |- ( ph -> RR C_ CC )
10		ssidd 3269	. . . . . . 7 |- ( ph -> RR C_ RR )
11	9, 1, 10	dvbss 15769	. . . . . 6 |- ( ph -> dom ( RR _D F ) C_ RR )
12		reldvg 15763	. . . . . . . . 9 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> Rel ( RR _D F ) )
13	9, 5, 12	syl2anc 415	. . . . . . . 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 15608	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
17		uniretop 15609	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
18	17	ntrtop 15212	. . . . . . . . . 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 2332	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` RR ) )
21		limcresi 15750	. . . . . . . . . 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 3269	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> CC C_ CC )
24		cncfmptc 15680	. . . . . . . . . . . 12 |- ( ( B e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
25	22, 8, 23, 24	mp3an12i 1382	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
26		eqidd 2239	. . . . . . . . . . 11 |- ( z = x -> B = B )
27	25, 15, 26	cnmptlimc 15758	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> B e. ( ( z e. RR |-> B ) lim CC x ) )
28	21, 27	sselid 3246	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> B e. ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x ) )
29		breq1 4131	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
30	29	elrab 2982	. . . . . . . . . . . . 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 1256	. . . . . . . . . . . . . 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 4219	. . . . . . . . . . 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 3333	. . . . . . . . . . . 12 |- { w e. RR | w # x } C_ RR
37		resmpt 5109	. . . . . . . . . . . 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 2289	. . . . . . . . . 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 6094	. . . . . . . . 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 2318	. . . . . . . 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 2238	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
43	42	tgioo2cntop 15641	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
44		eqid 2238	. . . . . . . . 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 3269	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> RR C_ RR )
48	43, 42, 44, 45, 46, 47	eldvap 15766	. . . . . . . 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 957	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> x ( RR _D F ) B )
50		releldm 5015	. . . . . . 7 |- ( ( Rel ( RR _D F ) /\ x ( RR _D F ) B ) -> x e. dom ( RR _D F ) )
51	14, 49, 50	syl2anc 415	. . . . . 6 |- ( ( ph /\ x e. RR ) -> x e. dom ( RR _D F ) )
52	11, 51	eqelssd 3267	. . . . 5 |- ( ph -> dom ( RR _D F ) = RR )
53	52	feq2d 5519	. . . 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 5532	. 2 |- ( ph -> ( RR _D F ) Fn RR )
56		fnconstg 5588	. . 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 5535	. . . . 5 |- ( ( ph /\ x e. RR ) -> Fun ( RR _D F ) )
60		funbrfvb 5740	. . . . 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 415	. . . 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 5923	. . . 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 2274	. 2 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) = ( ( RR X. { B } ) ` x ) )
67	55, 57, 66	eqfnfvd 5803	1 |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   {csn 3708   class class class wbr 4128   |-> cmpt 4190   X. cxp 4770   dom cdm 4772   ran crn 4773   |` cres 4774   o. ccom 4776   Rel wrel 4777   Fun wfun 5369   Fn wfn 5370   -->wf 5371   ` cfv 5375  (class class class)co 6079   ^pm cpm 6917   CCcc 8171   RRcr 8172   - cmin 8491   # cap 8903   / cdiv 8996   (,)cioo 10273   abscabs 11746   topGenctg 13591   MetOpencmopn 14861   Topctop 15081   intcnt 15177   -cn->ccncf 15654   lim CC climc 15738   _D cdv 15739

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-map 6918  df-pm 6919  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-xneg 10157  df-xadd 10158  df-ioo 10277  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-rest 13578  df-topgen 13597  df-psmet 14863  df-xmet 14864  df-met 14865  df-bl 14866  df-mopn 14867  df-top 15082  df-topon 15095  df-bases 15127  df-ntr 15180  df-cn 15272  df-cnp 15273  df-cncf 15655  df-limced 15740  df-dvap 15741

This theorem is referenced by:  dvconstre  15780  dvidre  15781