Theorem dvidrelem 14928

Description: Lemma for dvidre 14933 and dvconstre 14932. Analogue of dvidlemap 14927 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 8013	. . . . . . 7 |- RR e. _V
3		cnex 8003	. . . . . . 7 |- CC e. _V
4	2, 3	fpm 6740	. . . . . 6 |- ( F : RR --> CC -> F e. ( CC ^pm RR ) )
5	1, 4	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm RR ) )
6		dvfpm 14925	. . . . 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 7971	. . . . . . . 8 |- RR C_ CC
9	8	a1i 9	. . . . . . 7 |- ( ph -> RR C_ CC )
10		ssidd 3204	. . . . . . 7 |- ( ph -> RR C_ RR )
11	9, 1, 10	dvbss 14921	. . . . . 6 |- ( ph -> dom ( RR _D F ) C_ RR )
12		reldvg 14915	. . . . . . . . 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 14760	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
17		uniretop 14761	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
18	17	ntrtop 14364	. . . . . . . . . 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 2290	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` RR ) )
21		limcresi 14902	. . . . . . . . . 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 3204	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> CC C_ CC )
24		cncfmptc 14832	. . . . . . . . . . . 12 |- ( ( B e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
25	22, 8, 23, 24	mp3an12i 1352	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
26		eqidd 2197	. . . . . . . . . . 11 |- ( z = x -> B = B )
27	25, 15, 26	cnmptlimc 14910	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> B e. ( ( z e. RR |-> B ) lim CC x ) )
28	21, 27	sselid 3181	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> B e. ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x ) )
29		breq1 4036	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
30	29	elrab 2920	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 4123	. . . . . . . . . . 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 3268	. . . . . . . . . . . 12 |- { w e. RR | w # x } C_ RR
37		resmpt 4994	. . . . . . . . . . . 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 2247	. . . . . . . . . 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 5937	. . . . . . . . 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 2276	. . . . . . . 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 2196	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
43	42	tgioo2cntop 14793	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
44		eqid 2196	. . . . . . . . 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 3204	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> RR C_ RR )
48	43, 42, 44, 45, 46, 47	eldvap 14918	. . . . . . . 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 946	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> x ( RR _D F ) B )
50		releldm 4901	. . . . . . 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 3202	. . . . 5 |- ( ph -> dom ( RR _D F ) = RR )
53	52	feq2d 5395	. . . 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 5408	. 2 |- ( ph -> ( RR _D F ) Fn RR )
56		fnconstg 5455	. . 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 5411	. . . . 5 |- ( ( ph /\ x e. RR ) -> Fun ( RR _D F ) )
60		funbrfvb 5603	. . . . 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 5776	. . . 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 2232	. 2 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) = ( ( RR X. { B } ) ` x ) )
67	55, 57, 66	eqfnfvd 5662	1 |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   {csn 3622   class class class wbr 4033   |-> cmpt 4094   X. cxp 4661   dom cdm 4663   ran crn 4664   |` cres 4665   o. ccom 4667   Rel wrel 4668   Fun wfun 5252   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   ^pm cpm 6708   CCcc 7877   RRcr 7878   - cmin 8197   # cap 8608   / cdiv 8699   (,)cioo 9963   abscabs 11162   topGenctg 12925   MetOpencmopn 14097   Topctop 14233   intcnt 14329   -cn->ccncf 14806   lim CC climc 14890   _D cdv 14891

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pm 6710  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-ioo 9967  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  dvconstre  14932  dvidre  14933