Theorem dvidlem 19755

Description: Lemma for dvid 19757 and dvconst 19756. (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 19748	. . . 4 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3327	. . . . . . . 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 19741	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
6		reldv 19710	. . . . . . . . 9 |- Rel ( CC _D F )
7		simpr 448	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> x e. CC )
8		eqid 2404	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 18771	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. Top
10	8	cnfldtopon 18770	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	toponunii 16952	. . . . . . . . . . . . 13 |- CC = U. ( TopOpen ` ℂfld )
12	11	ntrtop 17089	. . . . . . . . . . . 12 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC )
13	9, 12	ax-mp 8	. . . . . . . . . . 11 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) = CC
14	7, 13	syl6eleqr 2495	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) )
15		limcresi 19725	. . . . . . . . . . . 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 18894	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
20	17, 18, 18, 19	syl3anc 1184	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
21		eqidd 2405	. . . . . . . . . . . . 13 |- ( z = x -> B = B )
22	20, 7, 21	cnmptlimc 19730	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
23	15, 22	sseldi 3306	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
24		eldifsn 3887	. . . . . . . . . . . . . . 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 1171	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z =/= x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
27	26	imp43 579	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z =/= x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
28	24, 27	sylan2b 462	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. CC ) /\ z e. ( CC \ { x } ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
29	28	mpteq2dva 4255	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> B ) )
30		difss 3434	. . . . . . . . . . . . . 14 |- ( CC \ { x } ) C_ CC
31		resmpt 5150	. . . . . . . . . . . . . 14 |- ( ( CC \ { x } ) C_ CC -> ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) = ( z e. ( CC \ { x } ) |-> B ) )
32	30, 31	ax-mp 8	. . . . . . . . . . . . 13 |- ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) = ( z e. ( CC \ { x } ) |-> B )
33	29, 32	syl6eqr 2454	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) )
34	33	oveq1d 6055	. . . . . . . . . . 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 2481	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
36	11	restid 13616	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
37	9, 36	ax-mp 8	. . . . . . . . . . . 12 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
38	37	eqcomi 2408	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
39		eqid 2404	. . . . . . . . . . 11 |- ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
40	4	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
41	38, 8, 39, 18, 40, 18	eldv 19738	. . . . . . . . . 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 889	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
43		releldm 5061	. . . . . . . . 9 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
44	6, 42, 43	sylancr 645	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
45	44	ex 424	. . . . . . 7 |- ( ph -> ( x e. CC -> x e. dom ( CC _D F ) ) )
46	45	ssrdv 3314	. . . . . 6 |- ( ph -> CC C_ dom ( CC _D F ) )
47	5, 46	eqssd 3325	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
48	47	feq2d 5540	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
49	1, 48	mpbii 203	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
50		ffn 5550	. . 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 5590	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	16, 52	mp1i 12	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54		ffun 5552	. . . . . 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 5728	. . . . 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 643	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	42, 57	mpbird 224	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	16	a1i 11	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 5904	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 458	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2439	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5789	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   \ cdif 3277   C_ wss 3280   {csn 3774   class class class wbr 4172   e. cmpt 4226   X. cxp 4835   dom cdm 4837   |` cres 4839   Rel wrel 4842   Fun wfun 5407   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   - cmin 9247   / cdiv 9633   ↾_t crest 13603   TopOpenctopn 13604  ℂ_fldccnfld 16658   Topctop 16913   intcnt 17036   -cn->ccncf 18859   lim CC climc 19702   _D cdv 19703

This theorem is referenced by:  dvconst  19756  dvid  19757

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-cncf 18861  df-limc 19706  df-dv 19707