Theorem dvidlem 22949

Description: Lemma for dvid 22951 and dvconst 22950. (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 22942	. . . 4 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3437	. . . . . . . 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 22935	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
6		reldv 22904	. . . . . . . . 9 |- Rel ( CC _D F )
7		simpr 468	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> x e. CC )
8		eqid 2471	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 21882	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. Top
10	8	cnfldtopon 21881	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	toponunii 20024	. . . . . . . . . . . . 13 |- CC = U. ( TopOpen ` ℂfld )
12	11	ntrtop 20163	. . . . . . . . . . . 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 2560	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( TopOpen ` ℂfld ) ) ` CC ) )
15		limcresi 22919	. . . . . . . . . . . 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 22021	. . . . . . . . . . . . . 14 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
20	17, 18, 18, 19	syl3anc 1292	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
21		eqidd 2472	. . . . . . . . . . . . 13 |- ( z = x -> B = B )
22	20, 7, 21	cnmptlimc 22924	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
23	15, 22	sseldi 3416	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) lim CC x ) )
24		eldifsn 4088	. . . . . . . . . . . . . . 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 1251	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z =/= x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
27	26	imp43 606	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z =/= x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
28	24, 27	sylan2b 483	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. CC ) /\ z e. ( CC \ { x } ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
29	28	mpteq2dva 4482	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> B ) )
30		difss 3549	. . . . . . . . . . . . . 14 |- ( CC \ { x } ) C_ CC
31		resmpt 5160	. . . . . . . . . . . . . 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 2523	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` ( CC \ { x } ) ) )
34	33	oveq1d 6323	. . . . . . . . . . 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 2552	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
36	11	restid 15410	. . . . . . . . . . . . 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 2471	. . . . . . . . . . 11 |- ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( CC \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
40	4	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
41	38, 8, 39, 18, 40, 18	eldv 22932	. . . . . . . . . 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 936	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
43		releldm 5073	. . . . . . . . 9 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
44	6, 42, 43	sylancr 676	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
45	44	ex 441	. . . . . . 7 |- ( ph -> ( x e. CC -> x e. dom ( CC _D F ) ) )
46	45	ssrdv 3424	. . . . . 6 |- ( ph -> CC C_ dom ( CC _D F ) )
47	5, 46	eqssd 3435	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
48	47	feq2d 5725	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
49	1, 48	mpbii 216	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
50		ffn 5739	. . 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 5784	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	16, 52	mp1i 13	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54		ffun 5742	. . . . . 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 5921	. . . . 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 673	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	42, 57	mpbird 240	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	16	a1i 11	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 6134	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 479	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2508	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5994	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   C_ wss 3390   {csn 3959   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   dom cdm 4839   |` cres 4841   Rel wrel 4844   Fun wfun 5583   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   - cmin 9880   / cdiv 10291   ↾_t crest 15397   TopOpenctopn 15398  ℂ_fldccnfld 19047   Topctop 19994   intcnt 20109   -cn->ccncf 21986   lim CC climc 22896   _D cdv 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-cncf 21988  df-limc 22900  df-dv 22901

This theorem is referenced by:  dvconst  22950  dvid  22951