Theorem dvidlem 22744

Description: Lemma for dvid 22746 and dvconst 22745. (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 22737	. . . 4 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3480	. . . . . . . 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 22730	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
6		reldv 22699	. . . . . . . . 9 |- Rel ( CC _D F )
7		simpr 462	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> x e. CC )
8		eqid 2420	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 21708	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. Top
10	8	cnfldtopon 21707	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	toponunii 19871	. . . . . . . . . . . . 13 |- CC = U. ( TopOpen ` ℂfld )
12	11	ntrtop 20010	. . . . . . . . . . . 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 2519	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) )
15		limcresi 22714	. . . . . . . . . . . 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 21832	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
20	17, 18, 18, 19	syl3anc 1264	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
21		eqidd 2421	. . . . . . . . . . . . 13 |- ( z = x -> B = B )
22	20, 7, 21	cnmptlimc 22719	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
23	15, 22	sseldi 3459	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
24		eldifsn 4119	. . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z =/= x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
27	26	imp43 598	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z =/= x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
28	24, 27	sylan2b 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. CC ) /\ z e. ( CC \ { x } ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
29	28	mpteq2dva 4503	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> B ) )
30		difss 3589	. . . . . . . . . . . . . 14 |- ( CC \ { x } ) C_ CC
31		resmpt 5165	. . . . . . . . . . . . . 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 2479	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) )
34	33	oveq1d 6311	. . . . . . . . . . 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 2511	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
36	11	restid 15284	. . . . . . . . . . . . 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 2433	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
39		eqid 2420	. . . . . . . . . . 11 |- ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
40	4	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
41	38, 8, 39, 18, 40, 18	eldv 22727	. . . . . . . . . 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 930	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
43		releldm 5078	. . . . . . . . 9 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
44	6, 42, 43	sylancr 667	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
45	44	ex 435	. . . . . . 7 |- ( ph -> ( x e. CC -> x e. dom ( CC _D F ) ) )
46	45	ssrdv 3467	. . . . . 6 |- ( ph -> CC C_ dom ( CC _D F ) )
47	5, 46	eqssd 3478	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
48	47	feq2d 5724	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
49	1, 48	mpbii 214	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
50		ffn 5737	. . 3 |- ( ( CC _D F ) : CC --> CC -> ( CC _D F ) Fn CC )
51	49, 50	syl 17	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5779	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	16, 52	mp1i 13	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54		ffun 5739	. . . . . 6 |- ( ( CC _D F ) : dom ( CC _D F ) --> CC -> Fun ( CC _D F ) )
55	1, 54	mp1i 13	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5914	. . . . 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 665	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	42, 57	mpbird 235	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	16	a1i 11	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 6124	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 473	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2464	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5985	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   =/= wne 2616   \ cdif 3430   C_ wss 3433   {csn 3993   class class class wbr 4417   |-> cmpt 4475   X. cxp 4843   dom cdm 4845   |` cres 4847   Rel wrel 4850   Fun wfun 5586   Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   CCcc 9526   - cmin 9849   / cdiv 10258   ↾_t crest 15271   TopOpenctopn 15272  ℂ_fldccnfld 18898   Topctop 19841   intcnt 19956   -cn->ccncf 21797   lim CC climc 22691   _D cdv 22692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fi 7922  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-icc 11631  df-fz 11772  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-plusg 15155  df-mulr 15156  df-starv 15157  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-rest 15273  df-topn 15274  df-topgen 15294  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-fbas 18895  df-fg 18896  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-ntr 19959  df-cls 19960  df-nei 20038  df-lp 20076  df-perf 20077  df-cn 20167  df-cnp 20168  df-haus 20255  df-fil 20785  df-fm 20877  df-flim 20878  df-flf 20879  df-xms 21259  df-ms 21260  df-cncf 21799  df-limc 22695  df-dv 22696

This theorem is referenced by:  dvconst  22745  dvid  22746