Theorem dvidlem 21395

Description: Lemma for dvid 21397 and dvconst 21396. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Hypotheses

Ref	Expression
dvidlem.1	|- ( ph -> F : CC --> CC )
dvidlem.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
dvidlem	|- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

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

Proof of Theorem dvidlem

Step	Hyp	Ref	Expression
1		dvfcn 21388	. . . 4 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3380	. . . . . . . 8 |- CC C_ CC
3	2	a1i 11	. . . . . . 7 |- ( ph -> CC C_ CC )
4		dvidlem.1	. . . . . . 7 |- ( ph -> F : CC --> CC )
5	3, 4, 3	dvbss 21381	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
6		reldv 21350	. . . . . . . . 9 |- Rel ( CC _D F )
7		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> x e. CC )
8		eqid 2443	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 20368	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. Top
10	8	cnfldtopon 20367	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	toponunii 18542	. . . . . . . . . . . . 13 |- CC = U. ( TopOpen ` ℂfld )
12	11	ntrtop 18679	. . . . . . . . . . . 12 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC )
13	9, 12	ax-mp 5	. . . . . . . . . . 11 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC
14	7, 13	syl6eleqr 2534	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) )
15		limcresi 21365	. . . . . . . . . . . 12 |- ( ( z e. CC |-> B ) lim CC x ) C_ ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x )
16		dvidlem.3	. . . . . . . . . . . . . . 15 |- B e. CC
17	16	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. CC ) -> B e. CC )
18	2	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. CC ) -> CC C_ CC )
19		cncfmptc 20492	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
20	17, 18, 18, 19	syl3anc 1218	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
21		eqidd 2444	. . . . . . . . . . . . 13 |- ( z = x -> B = B )
22	20, 7, 21	cnmptlimc 21370	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
23	15, 22	sseldi 3359	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
24		eldifsn 4005	. . . . . . . . . . . . . . 15 |- ( z e. ( CC \ { x } ) <-> ( z e. CC /\ z =/= x ) )
25		dvidlem.2	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( x e. CC /\ z e. CC /\ z =/= x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
26	25	3exp2 1205	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z =/= x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
27	26	imp43 595	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z =/= x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
28	24, 27	sylan2b 475	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. CC ) /\ z e. ( CC \ { x } ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
29	28	mpteq2dva 4383	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> B ) )
30		difss 3488	. . . . . . . . . . . . . 14 |- ( CC \ { x } ) C_ CC
31		resmpt 5161	. . . . . . . . . . . . . 14 |- ( ( CC \ { x } ) C_ CC -> ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) = ( z e. ( CC \ { x } ) |-> B ) )
32	30, 31	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) = ( z e. ( CC \ { x } ) |-> B )
33	29, 32	syl6eqr 2493	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) )
34	33	oveq1d 6111	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
35	23, 34	eleqtrrd 2520	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
36	11	restid 14377	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
37	9, 36	ax-mp 5	. . . . . . . . . . . 12 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
38	37	eqcomi 2447	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
39		eqid 2443	. . . . . . . . . . 11 |- ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
40	4	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
41	38, 8, 39, 18, 40, 18	eldv 21378	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> ( x ( CC _D F ) B <-> ( x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) /\ B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
42	14, 35, 41	mpbir2and 913	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
43		releldm 5077	. . . . . . . . 9 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
44	6, 42, 43	sylancr 663	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
45	44	ex 434	. . . . . . 7 |- ( ph -> ( x e. CC -> x e. dom ( CC _D F ) ) )
46	45	ssrdv 3367	. . . . . 6 |- ( ph -> CC C_ dom ( CC _D F ) )
47	5, 46	eqssd 3378	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
48	47	feq2d 5552	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
49	1, 48	mpbii 211	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
50		ffn 5564	. . 3 |- ( ( CC _D F ) : CC --> CC -> ( CC _D F ) Fn CC )
51	49, 50	syl 16	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5603	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	16, 52	mp1i 12	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54		ffun 5566	. . . . . 6 |- ( ( CC _D F ) : dom ( CC _D F ) --> CC -> Fun ( CC _D F ) )
55	1, 54	mp1i 12	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5739	. . . . 5 |- ( ( Fun ( CC _D F ) /\ x e. dom ( CC _D F ) ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
57	55, 44, 56	syl2anc 661	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	42, 57	mpbird 232	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	16	a1i 11	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 5936	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 471	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2478	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5805	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2611   \ cdif 3330   C_ wss 3333   {csn 3882   class class class wbr 4297   e. cmpt 4355   X. cxp 4843   dom cdm 4845   |` cres 4847   Rel wrel 4850   Fun wfun 5417   Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   - cmin 9600   / cdiv 9998   ↾_t crest 14364   TopOpenctopn 14365  ℂ_fldccnfld 17823   Topctop 18503   intcnt 18626   -cn->ccncf 20457   lim CC climc 21342   _D cdv 21343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-cncf 20459  df-limc 21346  df-dv 21347

This theorem is referenced by:  dvconst  21396  dvid  21397