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Theorem dvidlem 19265
Description: Lemma for dvid 19267 and dvconst 19266. (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
StepHypRef Expression
1 dvfcn 19258 . . . 4  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3197 . . . . . . . 8  |-  CC  C_  CC
32a1i 10 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
4 dvidlem.1 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
53, 4, 3dvbss 19251 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
6 reldv 19220 . . . . . . . . 9  |-  Rel  ( CC  _D  F )
7 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
8 eqid 2283 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 18293 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  Top
108cnfldtopon 18292 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110toponunii 16670 . . . . . . . . . . . . 13  |-  CC  =  U. ( TopOpen ` fld )
1211ntrtop 16807 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
139, 12ax-mp 8 . . . . . . . . . . 11  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
147, 13syl6eleqr 2374 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( TopOpen
` fld
) ) `  CC ) )
15 limcresi 19235 . . . . . . . . . . . 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
1716a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
182a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  CC )  ->  CC  C_  CC )
19 cncfmptc 18415 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2017, 18, 18, 19syl3anc 1182 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
21 eqidd 2284 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  B  =  B )
2220, 7, 21cnmptlimc 19240 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2315, 22sseldi 3178 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  ( CC  \  {
x } ) ) lim
CC  x ) )
24 eldifsn 3749 . . . . . . . . . . . . . . 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 )
26253exp2 1169 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z  =/=  x  ->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
2726imp43 578 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z  =/=  x ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2824, 27sylan2b 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  ( CC  \  {
x } ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2928mpteq2dva 4106 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B ) )
30 difss 3303 . . . . . . . . . . . . . 14  |-  ( CC 
\  { x }
)  C_  CC
31 resmpt 5000 . . . . . . . . . . . . . 14  |-  ( ( CC  \  { x } )  C_  CC  ->  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) )  =  ( z  e.  ( CC  \  { x } )  |->  B ) )
3230, 31ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B )
3329, 32syl6eqr 2333 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) ) )
3433oveq1d 5873 . . . . . . . . . . 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 ) )
3523, 34eleqtrrd 2360 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  ( CC  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )
3611restid 13338 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
379, 36ax-mp 8 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3837eqcomi 2287 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
39 eqid 2283 . . . . . . . . . . 11  |-  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
404adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4138, 8, 39, 18, 40, 18eldv 19248 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  ( x ( CC  _D  F
) B  <->  ( x  e.  ( ( int `  ( TopOpen
` fld
) ) `  CC )  /\  B  e.  ( ( z  e.  ( CC  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) ) )
4214, 35, 41mpbir2and 888 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
43 releldm 4911 . . . . . . . . 9  |-  ( ( Rel  ( CC  _D  F )  /\  x
( CC  _D  F
) B )  ->  x  e.  dom  ( CC 
_D  F ) )
446, 42, 43sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e. 
dom  ( CC  _D  F ) )
4544ex 423 . . . . . . 7  |-  ( ph  ->  ( x  e.  CC  ->  x  e.  dom  ( CC  _D  F ) ) )
4645ssrdv 3185 . . . . . 6  |-  ( ph  ->  CC  C_  dom  ( CC 
_D  F ) )
475, 46eqssd 3196 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4847feq2d 5380 . . . 4  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : CC --> CC ) )
491, 48mpbii 202 . . 3  |-  ( ph  ->  ( CC  _D  F
) : CC --> CC )
50 ffn 5389 . . 3  |-  ( ( CC  _D  F ) : CC --> CC  ->  ( CC  _D  F )  Fn  CC )
5149, 50syl 15 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5429 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5316, 52mp1i 11 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
54 ffun 5391 . . . . . 6  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
551, 54mp1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  Fun  ( CC  _D  F ) )
56 funbrfvb 5565 . . . . 5  |-  ( ( Fun  ( CC  _D  F )  /\  x  e.  dom  ( CC  _D  F ) )  -> 
( ( ( CC 
_D  F ) `  x )  =  B  <-> 
x ( CC  _D  F ) B ) )
5755, 44, 56syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( ( CC  _D  F
) `  x )  =  B  <->  x ( CC 
_D  F ) B ) )
5842, 57mpbird 223 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  B )
5916a1i 10 . . . 4  |-  ( ph  ->  B  e.  CC )
60 fvconst2g 5727 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 457 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2318 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5625 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    - cmin 9037    / cdiv 9423   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631   intcnt 16754   -cn->ccncf 18380   lim CC climc 19212    _D cdv 19213
This theorem is referenced by:  dvconst  19266  dvid  19267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-cncf 18382  df-limc 19216  df-dv 19217
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