Theorem dvidlemap 15163

Description: Lemma for dvid 15167 and dvconst 15166. (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 8049	. . . . . . 7 |- CC e. _V
3	2, 2	fpm 6768	. . . . . 6 |- ( F : CC --> CC -> F e. ( CC ^pm CC ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm CC ) )
5		dvfcnpm 15162	. . . . 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 3214	. . . . . . 7 |- ( ph -> CC C_ CC )
8	7, 1, 7	dvbss 15157	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
9		reldvg 15151	. . . . . . . . 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 2205	. . . . . . . . . . 11 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
14	13	cntoptop 15005	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) e. Top
15	13	cntoptopon 15004	. . . . . . . . . . . 12 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
16	15	toponunii 14489	. . . . . . . . . . 11 |- CC = U. ( MetOpen ` ( abs o. - ) )
17	16	ntrtop 14600	. . . . . . . . . 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 2299	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) )
20		limcresi 15138	. . . . . . . . . 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 3214	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> CC C_ CC )
23		cncfmptc 15068	. . . . . . . . . . . 12 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
24	21, 22, 22, 23	mp3an2i 1355	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
25		eqidd 2206	. . . . . . . . . . 11 |- ( z = x -> B = B )
26	24, 12, 25	cnmptlimc 15146	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
27	20, 26	sselid 3191	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
28		breq1 4047	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
29	28	elrab 2929	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 4134	. . . . . . . . . . 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 3278	. . . . . . . . . . . 12 |- { w e. CC | w # x } C_ CC
36		resmpt 5007	. . . . . . . . . . . 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 2256	. . . . . . . . . 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 5959	. . . . . . . . 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 2285	. . . . . . . 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 14493	. . . . . . . . 9 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
42		eqid 2205	. . . . . . . . 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 15154	. . . . . . . 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 947	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
46		releldm 4913	. . . . . . 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 3212	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
49	48	feq2d 5413	. . . 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 5426	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5473	. . 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 5429	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5621	. . . . 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 5798	. . . 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 2241	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5680	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   C_ wss 3166   {csn 3633   class class class wbr 4044   |-> cmpt 4105   X. cxp 4673   dom cdm 4675   |` cres 4677   o. ccom 4679   Rel wrel 4680   Fun wfun 5265   Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5944   ^pm cpm 6736   CCcc 7923   - cmin 8243   # cap 8654   / cdiv 8745   abscabs 11308   MetOpencmopn 14303   Topctop 14469   intcnt 14565   -cn->ccncf 15042   lim CC climc 15126   _D cdv 15127

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-map 6737  df-pm 6738  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-xneg 9894  df-xadd 9895  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-rest 13073  df-topgen 13092  df-psmet 14305  df-xmet 14306  df-met 14307  df-bl 14308  df-mopn 14309  df-top 14470  df-topon 14483  df-bases 14515  df-ntr 14568  df-cn 14660  df-cnp 14661  df-cncf 15043  df-limced 15128  df-dvap 15129

This theorem is referenced by:  dvconst  15166  dvid  15167