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Theorem dvidlem 19281
Description: Lemma for dvid 19283 and dvconst 19282. (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 19274 . . . 4  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3210 . . . . . . . 8  |-  CC  C_  CC
32a1i 10 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
4 dvidlem.1 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
53, 4, 3dvbss 19267 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
6 reldv 19236 . . . . . . . . 9  |-  Rel  ( CC  _D  F )
7 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
8 eqid 2296 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 18309 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  Top
108cnfldtopon 18308 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110toponunii 16686 . . . . . . . . . . . . 13  |-  CC  =  U. ( TopOpen ` fld )
1211ntrtop 16823 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
139, 12ax-mp 8 . . . . . . . . . . 11  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
147, 13syl6eleqr 2387 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( TopOpen
` fld
) ) `  CC ) )
15 limcresi 19251 . . . . . . . . . . . 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 18431 . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  B  =  B )
2220, 7, 21cnmptlimc 19256 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2315, 22sseldi 3191 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  ( CC  \  {
x } ) ) lim
CC  x ) )
24 eldifsn 3762 . . . . . . . . . . . . . . 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 4122 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B ) )
30 difss 3316 . . . . . . . . . . . . . 14  |-  ( CC 
\  { x }
)  C_  CC
31 resmpt 5016 . . . . . . . . . . . . . 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 2346 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) ) )
3433oveq1d 5889 . . . . . . . . . . 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 2373 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  ( CC  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )
3611restid 13354 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
379, 36ax-mp 8 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3837eqcomi 2300 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
39 eqid 2296 . . . . . . . . . . 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 19264 . . . . . . . . . 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 4927 . . . . . . . . 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 3198 . . . . . 6  |-  ( ph  ->  CC  C_  dom  ( CC 
_D  F ) )
475, 46eqssd 3209 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4847feq2d 5396 . . . 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 5405 . . 3  |-  ( ( CC  _D  F ) : CC --> CC  ->  ( CC  _D  F )  Fn  CC )
5149, 50syl 15 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5445 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5316, 52mp1i 11 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
54 ffun 5407 . . . . . 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 5581 . . . . 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 5743 . . . 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 2331 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5641 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    - cmin 9053    / cdiv 9439   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647   intcnt 16770   -cn->ccncf 18396   lim CC climc 19228    _D cdv 19229
This theorem is referenced by:  dvconst  19282  dvid  19283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-cncf 18398  df-limc 19232  df-dv 19233
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