Theorem dvidlemap 12829

Description: Lemma for dvid 12831 and dvconst 12830. (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 7744	. . . . . . 7 |- CC e. _V
3	2, 2	fpm 6575	. . . . . 6 |- ( F : CC --> CC -> F e. ( CC ^pm CC ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm CC ) )
5		dvfcnpm 12828	. . . . 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 3118	. . . . . . 7 |- ( ph -> CC C_ CC )
8	7, 1, 7	dvbss 12823	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
9		reldvg 12817	. . . . . . . . 9 |- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) ) -> Rel ( CC _D F ) )
10	7, 4, 9	syl2anc 408	. . . . . . . 8 |- ( ph -> Rel ( CC _D F ) )
11	10	adantr 274	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> Rel ( CC _D F ) )
12		simpr 109	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x e. CC )
13		eqid 2139	. . . . . . . . . . 11 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
14	13	cntoptop 12702	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) e. Top
15	13	cntoptopon 12701	. . . . . . . . . . . 12 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
16	15	toponunii 12184	. . . . . . . . . . 11 |- CC = U. ( MetOpen ` ( abs o. - ) )
17	16	ntrtop 12297	. . . . . . . . . 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 2233	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) )
20		limcresi 12804	. . . . . . . . . 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 3118	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> CC C_ CC )
23		cncfmptc 12751	. . . . . . . . . . . 12 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
24	21, 22, 22, 23	mp3an2i 1320	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
25		eqidd 2140	. . . . . . . . . . 11 |- ( z = x -> B = B )
26	24, 12, 25	cnmptlimc 12812	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
27	20, 26	sseldi 3095	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
28		breq1 3932	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
29	28	elrab 2840	. . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
32	31	imp43 352	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
33	29, 32	sylan2b 285	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. CC ) /\ z e. { w e. CC | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
34	33	mpteq2dva 4018	. . . . . . . . . . 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 3182	. . . . . . . . . . . 12 |- { w e. CC | w # x } C_ CC
36		resmpt 4867	. . . . . . . . . . . 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	syl6eqr 2190	. . . . . . . . . 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 5789	. . . . . . . . 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 2219	. . . . . . . 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 12188	. . . . . . . . 9 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
42		eqid 2139	. . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
44	41, 13, 42, 22, 43, 22	eldvap 12820	. . . . . . . 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 928	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
46		releldm 4774	. . . . . . 7 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
47	11, 45, 46	syl2anc 408	. . . . . 6 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
48	8, 47	eqelssd 3116	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
49	48	feq2d 5260	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
50	6, 49	mpbid 146	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
51	50	ffnd 5273	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5320	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	21, 52	mp1i 10	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54	6	adantr 274	. . . . . 6 |- ( ( ph /\ x e. CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
55	54	ffund 5276	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5464	. . . . 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 408	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	45, 57	mpbird 166	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	21	a1i 9	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 5634	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 281	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2175	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5521	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2420   C_ wss 3071   {csn 3527   class class class wbr 3929   |-> cmpt 3989   X. cxp 4537   dom cdm 4539   |` cres 4541   o. ccom 4543   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   ^pm cpm 6543   CCcc 7618   - cmin 7933   # cap 8343   / cdiv 8432   abscabs 10769   MetOpencmopn 12154   Topctop 12164   intcnt 12262   -cn->ccncf 12726   lim CC climc 12792   _D cdv 12793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-cncf 12727  df-limced 12794  df-dvap 12795

This theorem is referenced by:  dvconst  12830  dvid  12831