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Theorem dvidlem 21349
Description: Lemma for dvid 21351 and dvconst 21350. (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 21342 . . . 4  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3372 . . . . . . . 8  |-  CC  C_  CC
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
4 dvidlem.1 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
53, 4, 3dvbss 21335 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
6 reldv 21304 . . . . . . . . 9  |-  Rel  ( CC  _D  F )
7 simpr 458 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
8 eqid 2441 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 20322 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  Top
108cnfldtopon 20321 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110toponunii 18496 . . . . . . . . . . . . 13  |-  CC  =  U. ( TopOpen ` fld )
1211ntrtop 18633 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
139, 12ax-mp 5 . . . . . . . . . . 11  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
147, 13syl6eleqr 2532 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( TopOpen
` fld
) ) `  CC ) )
15 limcresi 21319 . . . . . . . . . . . 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 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
182a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  CC )  ->  CC  C_  CC )
19 cncfmptc 20446 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2017, 18, 18, 19syl3anc 1213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
21 eqidd 2442 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  B  =  B )
2220, 7, 21cnmptlimc 21324 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2315, 22sseldi 3351 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  ( CC  \  {
x } ) ) lim
CC  x ) )
24 eldifsn 3997 . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z  =/=  x  ->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
2726imp43 592 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z  =/=  x ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2824, 27sylan2b 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  ( CC  \  {
x } ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2928mpteq2dva 4375 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B ) )
30 difss 3480 . . . . . . . . . . . . . 14  |-  ( CC 
\  { x }
)  C_  CC
31 resmpt 5153 . . . . . . . . . . . . . 14  |-  ( ( CC  \  { x } )  C_  CC  ->  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) )  =  ( z  e.  ( CC  \  { x } )  |->  B ) )
3230, 31ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B )
3329, 32syl6eqr 2491 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) ) )
3433oveq1d 6105 . . . . . . . . . . 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 2518 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  ( CC  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )
3611restid 14368 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
379, 36ax-mp 5 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3837eqcomi 2445 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
39 eqid 2441 . . . . . . . . . . 11  |-  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
404adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4138, 8, 39, 18, 40, 18eldv 21332 . . . . . . . . . 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 908 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
43 releldm 5068 . . . . . . . . 9  |-  ( ( Rel  ( CC  _D  F )  /\  x
( CC  _D  F
) B )  ->  x  e.  dom  ( CC 
_D  F ) )
446, 42, 43sylancr 658 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e. 
dom  ( CC  _D  F ) )
4544ex 434 . . . . . . 7  |-  ( ph  ->  ( x  e.  CC  ->  x  e.  dom  ( CC  _D  F ) ) )
4645ssrdv 3359 . . . . . 6  |-  ( ph  ->  CC  C_  dom  ( CC 
_D  F ) )
475, 46eqssd 3370 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4847feq2d 5544 . . . 4  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : CC --> CC ) )
491, 48mpbii 211 . . 3  |-  ( ph  ->  ( CC  _D  F
) : CC --> CC )
50 ffn 5556 . . 3  |-  ( ( CC  _D  F ) : CC --> CC  ->  ( CC  _D  F )  Fn  CC )
5149, 50syl 16 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5595 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5316, 52mp1i 12 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
54 ffun 5558 . . . . . 6  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
551, 54mp1i 12 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  Fun  ( CC  _D  F ) )
56 funbrfvb 5731 . . . . 5  |-  ( ( Fun  ( CC  _D  F )  /\  x  e.  dom  ( CC  _D  F ) )  -> 
( ( ( CC 
_D  F ) `  x )  =  B  <-> 
x ( CC  _D  F ) B ) )
5755, 44, 56syl2anc 656 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( ( CC  _D  F
) `  x )  =  B  <->  x ( CC 
_D  F ) B ) )
5842, 57mpbird 232 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  B )
5916a1i 11 . . . 4  |-  ( ph  ->  B  e.  CC )
60 fvconst2g 5928 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 468 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2476 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5797 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322    C_ wss 3325   {csn 3874   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   dom cdm 4836    |` cres 4838   Rel wrel 4841   Fun wfun 5409    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276    - cmin 9591    / cdiv 9989   ↾t crest 14355   TopOpenctopn 14356  ℂfldccnfld 17777   Topctop 18457   intcnt 18580   -cn->ccncf 20411   lim CC climc 21296    _D cdv 21297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-rest 14357  df-topn 14358  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-cncf 20413  df-limc 21300  df-dv 21301
This theorem is referenced by:  dvconst  21350  dvid  21351
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