MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvidlem Structured version   Visualization version   Unicode version

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
StepHypRef Expression
1 dvfcn 22942 . . . 4  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3437 . . . . . . . 8  |-  CC  C_  CC
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
4 dvidlem.1 . . . . . . 7  |-  ( ph  ->  F : CC --> CC )
53, 4, 3dvbss 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 ` fld )  =  ( TopOpen ` fld )
98cnfldtop 21882 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  Top
108cnfldtopon 21881 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110toponunii 20024 . . . . . . . . . . . . 13  |-  CC  =  U. ( TopOpen ` fld )
1211ntrtop 20163 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
139, 12ax-mp 5 . . . . . . . . . . 11  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
147, 13syl6eleqr 2560 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( TopOpen
` fld
) ) `  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
1716a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
182a1i 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 ) )
2017, 18, 18, 19syl3anc 1292 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
21 eqidd 2472 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  B  =  B )
2220, 7, 21cnmptlimc 22924 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2315, 22sseldi 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 )
26253exp2 1251 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z  =/=  x  ->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
2726imp43 606 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z  =/=  x ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2824, 27sylan2b 483 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  ( CC  \  {
x } ) )  ->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) )  =  B )
2928mpteq2dva 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 ) )
3230, 31ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) )  =  ( z  e.  ( CC  \  { x } ) 
|->  B )
3329, 32syl6eqr 2523 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  ( CC  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( ( z  e.  CC  |->  B )  |`  ( CC  \  { x } ) ) )
3433oveq1d 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 ) )
3523, 34eleqtrrd 2552 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  ( CC  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )
3611restid 15410 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
379, 36ax-mp 5 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
3837eqcomi 2480 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )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 ) ) )
404adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4138, 8, 39, 18, 40, 18eldv 22932 . . . . . . . . . 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 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 ) )
446, 42, 43sylancr 676 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e. 
dom  ( CC  _D  F ) )
4544ex 441 . . . . . . 7  |-  ( ph  ->  ( x  e.  CC  ->  x  e.  dom  ( CC  _D  F ) ) )
4645ssrdv 3424 . . . . . 6  |-  ( ph  ->  CC  C_  dom  ( CC 
_D  F ) )
475, 46eqssd 3435 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4847feq2d 5725 . . . 4  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : CC --> CC ) )
491, 48mpbii 216 . . 3  |-  ( ph  ->  ( CC  _D  F
) : CC --> CC )
50 ffn 5739 . . 3  |-  ( ( CC  _D  F ) : CC --> CC  ->  ( CC  _D  F )  Fn  CC )
5149, 50syl 17 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5784 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5316, 52mp1i 13 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
54 ffun 5742 . . . . . 6  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
551, 54mp1i 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 ) )
5755, 44, 56syl2anc 673 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( ( CC  _D  F
) `  x )  =  B  <->  x ( CC 
_D  F ) B ) )
5842, 57mpbird 240 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  B )
5916a1i 11 . . . 4  |-  ( ph  ->  B  e.  CC )
60 fvconst2g 6134 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 479 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2508 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5994 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff setvar class
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  ℂfldccnfld 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
  Copyright terms: Public domain W3C validator