Theorem dvidrelem 15549

Description: Lemma for dvidre 15554 and dvconstre 15553. Analogue of dvidlemap 15548 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 8260	. . . . . . 7 |- RR e. _V
3		cnex 8250	. . . . . . 7 |- CC e. _V
4	2, 3	fpm 6914	. . . . . 6 |- ( F : RR --> CC -> F e. ( CC ^pm RR ) )
5	1, 4	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm RR ) )
6		dvfpm 15546	. . . . 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 8218	. . . . . . . 8 |- RR C_ CC
9	8	a1i 9	. . . . . . 7 |- ( ph -> RR C_ CC )
10		ssidd 3258	. . . . . . 7 |- ( ph -> RR C_ RR )
11	9, 1, 10	dvbss 15542	. . . . . 6 |- ( ph -> dom ( RR _D F ) C_ RR )
12		reldvg 15536	. . . . . . . . 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 15381	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
17		uniretop 15382	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
18	17	ntrtop 14985	. . . . . . . . . 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 2326	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> x e. ( ( int ` ( topGen ` ran (,) ) ) ` RR ) )
21		limcresi 15523	. . . . . . . . . 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 3258	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> CC C_ CC )
24		cncfmptc 15453	. . . . . . . . . . . 12 |- ( ( B e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
25	22, 8, 23, 24	mp3an12i 1378	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( z e. RR |-> B ) e. ( RR -cn-> CC ) )
26		eqidd 2233	. . . . . . . . . . 11 |- ( z = x -> B = B )
27	25, 15, 26	cnmptlimc 15531	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> B e. ( ( z e. RR |-> B ) lim CC x ) )
28	21, 27	sselid 3235	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> B e. ( ( ( z e. RR |-> B ) |` { w e. RR | w # x } ) lim CC x ) )
29		breq1 4111	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
30	29	elrab 2972	. . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . 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 4199	. . . . . . . . . . 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 3322	. . . . . . . . . . . 12 |- { w e. RR | w # x } C_ RR
37		resmpt 5085	. . . . . . . . . . . 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 2283	. . . . . . . . . 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 6064	. . . . . . . . 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 2312	. . . . . . . 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 2232	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
43	42	tgioo2cntop 15414	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
44		eqid 2232	. . . . . . . . 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 3258	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> RR C_ RR )
48	43, 42, 44, 45, 46, 47	eldvap 15539	. . . . . . . 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 953	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> x ( RR _D F ) B )
50		releldm 4991	. . . . . . 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 3256	. . . . 5 |- ( ph -> dom ( RR _D F ) = RR )
53	52	feq2d 5495	. . . 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 5508	. 2 |- ( ph -> ( RR _D F ) Fn RR )
56		fnconstg 5564	. . 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 5511	. . . . 5 |- ( ( ph /\ x e. RR ) -> Fun ( RR _D F ) )
60		funbrfvb 5716	. . . . 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 5897	. . . 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 2268	. 2 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) = ( ( RR X. { B } ) ` x ) )
67	55, 57, 66	eqfnfvd 5777	1 |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3210   {csn 3688   class class class wbr 4108   |-> cmpt 4170   X. cxp 4746   dom cdm 4748   ran crn 4749   |` cres 4750   o. ccom 4752   Rel wrel 4753   Fun wfun 5345   Fn wfn 5346   -->wf 5347   ` cfv 5351  (class class class)co 6049   ^pm cpm 6882   CCcc 8124   RRcr 8125   - cmin 8443   # cap 8854   / cdiv 8945   (,)cioo 10220   abscabs 11678   topGenctg 13459   MetOpencmopn 14681   Topctop 14854   intcnt 14950   -cn->ccncf 15427   lim CC climc 15511   _D cdv 15512

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 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-stab 839  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-map 6883  df-pm 6884  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-xneg 10104  df-xadd 10105  df-ioo 10224  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-rest 13446  df-topgen 13465  df-psmet 14683  df-xmet 14684  df-met 14685  df-bl 14686  df-mopn 14687  df-top 14855  df-topon 14868  df-bases 14900  df-ntr 14953  df-cn 15045  df-cnp 15046  df-cncf 15428  df-limced 15513  df-dvap 15514

This theorem is referenced by:  dvconstre  15553  dvidre  15554