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Theorem dvidlem 21395
Description: Lemma for dvid 21397 and dvconst 21396. (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 21388 . . . 4  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3380 . . . . . . . 8  |-  CC  C_  CC
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
4 dvidlem.1 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
53, 4, 3dvbss 21381 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
6 reldv 21350 . . . . . . . . 9  |-  Rel  ( CC  _D  F )
7 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
8 eqid 2443 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 20368 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  Top
108cnfldtopon 20367 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110toponunii 18542 . . . . . . . . . . . . 13  |-  CC  =  U. ( TopOpen ` fld )
1211ntrtop 18679 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
139, 12ax-mp 5 . . . . . . . . . . 11  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
147, 13syl6eleqr 2534 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( TopOpen
` fld
) ) `  CC ) )
15 limcresi 21365 . . . . . . . . . . . 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 20492 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2017, 18, 18, 19syl3anc 1218 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
21 eqidd 2444 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  B  =  B )
2220, 7, 21cnmptlimc 21370 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2315, 22sseldi 3359 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  ( CC  \  {
x } ) ) lim
CC  x ) )
24 eldifsn 4005 . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z  =/=  x  ->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
2726imp43 595 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z  =/=  x ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2824, 27sylan2b 475 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  ( CC  \  {
x } ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2928mpteq2dva 4383 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B ) )
30 difss 3488 . . . . . . . . . . . . . 14  |-  ( CC 
\  { x }
)  C_  CC
31 resmpt 5161 . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) ) )
3433oveq1d 6111 . . . . . . . . . . 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 2520 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  ( CC  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )
3611restid 14377 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
379, 36ax-mp 5 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3837eqcomi 2447 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
39 eqid 2443 . . . . . . . . . . 11  |-  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
404adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4138, 8, 39, 18, 40, 18eldv 21378 . . . . . . . . . 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 913 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
43 releldm 5077 . . . . . . . . 9  |-  ( ( Rel  ( CC  _D  F )  /\  x
( CC  _D  F
) B )  ->  x  e.  dom  ( CC 
_D  F ) )
446, 42, 43sylancr 663 . . . . . . . 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 3367 . . . . . 6  |-  ( ph  ->  CC  C_  dom  ( CC 
_D  F ) )
475, 46eqssd 3378 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4847feq2d 5552 . . . 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 5564 . . 3  |-  ( ( CC  _D  F ) : CC --> CC  ->  ( CC  _D  F )  Fn  CC )
5149, 50syl 16 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5603 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5316, 52mp1i 12 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
54 ffun 5566 . . . . . 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 5739 . . . . 5  |-  ( ( Fun  ( CC  _D  F )  /\  x  e.  dom  ( CC  _D  F ) )  -> 
( ( ( CC 
_D  F ) `  x )  =  B  <-> 
x ( CC  _D  F ) B ) )
5755, 44, 56syl2anc 661 . . . 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 5936 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 471 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2478 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5805 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330    C_ wss 3333   {csn 3882   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   dom cdm 4845    |` cres 4847   Rel wrel 4850   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285    - cmin 9600    / cdiv 9998   ↾t crest 14364   TopOpenctopn 14365  ℂfldccnfld 17823   Topctop 18503   intcnt 18626   -cn->ccncf 20457   lim CC climc 21342    _D cdv 21343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-cncf 20459  df-limc 21346  df-dv 21347
This theorem is referenced by:  dvconst  21396  dvid  21397
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