Theorem dvidlemap 15556

Description: Lemma for dvid 15560 and dvconst 15559. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)

Hypotheses

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

Assertion

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

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

Proof of Theorem dvidlemap

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvidlem.1	. . . . . 6 |- ( ph -> F : CC --> CC )
2		cnex 8251	. . . . . . 7 |- CC e. _V
3	2, 2	fpm 6915	. . . . . 6 |- ( F : CC --> CC -> F e. ( CC ^pm CC ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm CC ) )
5		dvfcnpm 15555	. . . . 5 |- ( F e. ( CC ^pm CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
6	4, 5	syl 14	. . . 4 |- ( ph -> ( CC _D F ) : dom ( CC _D F ) --> CC )
7		ssidd 3259	. . . . . . 7 |- ( ph -> CC C_ CC )
8	7, 1, 7	dvbss 15550	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
9		reldvg 15544	. . . . . . . . 9 |- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) ) -> Rel ( CC _D F ) )
10	7, 4, 9	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( CC _D F ) )
11	10	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> Rel ( CC _D F ) )
12		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x e. CC )
13		eqid 2232	. . . . . . . . . . 11 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
14	13	cntoptop 15398	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) e. Top
15	13	cntoptopon 15397	. . . . . . . . . . . 12 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
16	15	toponunii 14882	. . . . . . . . . . 11 |- CC = U. ( MetOpen ` ( abs o. - ) )
17	16	ntrtop 14993	. . . . . . . . . 10 |- ( ( MetOpen ` ( abs o. - ) ) e. Top -> ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) = CC )
18	14, 17	ax-mp 5	. . . . . . . . 9 |- ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) = CC
19	12, 18	eleqtrrdi 2326	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) )
20		limcresi 15531	. . . . . . . . . 10 |- ( ( z e. CC |-> B ) lim CC x ) C_ ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x )
21		dvidlem.3	. . . . . . . . . . . 12 |- B e. CC
22		ssidd 3259	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> CC C_ CC )
23		cncfmptc 15461	. . . . . . . . . . . 12 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
24	21, 22, 22, 23	mp3an2i 1379	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
25		eqidd 2233	. . . . . . . . . . 11 |- ( z = x -> B = B )
26	24, 12, 25	cnmptlimc 15539	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
27	20, 26	sselid 3236	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
28		breq1 4112	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
29	28	elrab 2973	. . . . . . . . . . . . 13 |- ( z e. { w e. CC | w # x } <-> ( z e. CC /\ z # x ) )
30		dvidlemap.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. CC /\ z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
31	30	3exp2 1252	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
32	31	imp43 355	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
33	29, 32	sylan2b 287	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. CC ) /\ z e. { w e. CC | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
34	33	mpteq2dva 4200	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. CC | w # x } |-> B ) )
35		ssrab2 3323	. . . . . . . . . . . 12 |- { w e. CC | w # x } C_ CC
36		resmpt 5086	. . . . . . . . . . . 12 |- ( { w e. CC | w # x } C_ CC -> ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) = ( z e. { w e. CC | w # x } |-> B ) )
37	35, 36	ax-mp 5	. . . . . . . . . . 11 |- ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) = ( z e. { w e. CC | w # x } |-> B )
38	34, 37	eqtr4di 2283	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) )
39	38	oveq1d 6065	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
40	27, 39	eleqtrrd 2312	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
41	15	toponrestid 14886	. . . . . . . . 9 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
42		eqid 2232	. . . . . . . . 9 |- ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
43	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
44	41, 13, 42, 22, 43, 22	eldvap 15547	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( x ( CC _D F ) B <-> ( x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) /\ B e. ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
45	19, 40, 44	mpbir2and 953	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
46		releldm 4992	. . . . . . 7 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
47	11, 45, 46	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
48	8, 47	eqelssd 3257	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
49	48	feq2d 5496	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
50	6, 49	mpbid 147	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
51	50	ffnd 5509	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5565	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	21, 52	mp1i 10	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54	6	adantr 276	. . . . . 6 |- ( ( ph /\ x e. CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
55	54	ffund 5512	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5717	. . . . 5 |- ( ( Fun ( CC _D F ) /\ x e. dom ( CC _D F ) ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
57	55, 47, 56	syl2anc 411	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	45, 57	mpbird 167	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	21	a1i 9	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 5898	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 283	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2268	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5778	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3211   {csn 3689   class class class wbr 4109   |-> cmpt 4171   X. cxp 4747   dom cdm 4749   |` cres 4751   o. ccom 4753   Rel wrel 4754   Fun wfun 5346   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   ^pm cpm 6883   CCcc 8125   - cmin 8444   # cap 8855   / cdiv 8946   abscabs 11682   MetOpencmopn 14689   Topctop 14862   intcnt 14958   -cn->ccncf 15435   lim CC climc 15519   _D cdv 15520

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 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-pm 6885  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-xneg 10105  df-xadd 10106  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-rest 13454  df-topgen 13473  df-psmet 14691  df-xmet 14692  df-met 14693  df-bl 14694  df-mopn 14695  df-top 14863  df-topon 14876  df-bases 14908  df-ntr 14961  df-cn 15053  df-cnp 15054  df-cncf 15436  df-limced 15521  df-dvap 15522

This theorem is referenced by:  dvconst  15559  dvid  15560