Theorem dvidlem 22054

Description: Lemma for dvid 22056 and dvconst 22055. (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 22047	. . . 4 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3523	. . . . . . . 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 22040	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
6		reldv 22009	. . . . . . . . 9 |- Rel ( CC _D F )
7		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> x e. CC )
8		eqid 2467	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 21026	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. Top
10	8	cnfldtopon 21025	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	toponunii 19200	. . . . . . . . . . . . 13 |- CC = U. ( TopOpen ` ℂfld )
12	11	ntrtop 19337	. . . . . . . . . . . 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 2566	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) )
15		limcresi 22024	. . . . . . . . . . . 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 21150	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
20	17, 18, 18, 19	syl3anc 1228	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
21		eqidd 2468	. . . . . . . . . . . . 13 |- ( z = x -> B = B )
22	20, 7, 21	cnmptlimc 22029	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
23	15, 22	sseldi 3502	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
24		eldifsn 4152	. . . . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . 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 4533	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> B ) )
30		difss 3631	. . . . . . . . . . . . . 14 |- ( CC \ { x } ) C_ CC
31		resmpt 5321	. . . . . . . . . . . . . 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 2526	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) )
34	33	oveq1d 6297	. . . . . . . . . . 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 2558	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
36	11	restid 14685	. . . . . . . . . . . . 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 2480	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
39		eqid 2467	. . . . . . . . . . 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 22037	. . . . . . . . . 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 920	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
43		releldm 5233	. . . . . . . . 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 3510	. . . . . 6 |- ( ph -> CC C_ dom ( CC _D F ) )
47	5, 46	eqssd 3521	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
48	47	feq2d 5716	. . . 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 5729	. . 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 5771	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	16, 52	mp1i 12	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54		ffun 5731	. . . . . 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 5908	. . . . 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 6112	. . . 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 2511	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5976	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   C_ wss 3476   {csn 4027   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   dom cdm 4999   |` cres 5001   Rel wrel 5004   Fun wfun 5580   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   - cmin 9801   / cdiv 10202   ↾_t crest 14672   TopOpenctopn 14673  ℂ_fldccnfld 18191   Topctop 19161   intcnt 19284   -cn->ccncf 21115   lim CC climc 22001   _D cdv 22002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-cncf 21117  df-limc 22005  df-dv 22006

This theorem is referenced by:  dvconst  22055  dvid  22056