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Theorem dvidlemap 12615
Description: Lemma for dvid 12617 and dvconst 12616. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
Hypotheses
Ref Expression
dvidlem.1  |-  ( ph  ->  F : CC --> CC )
dvidlemap.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
dvidlemap  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Distinct variable groups:    x, z, B   
x, F, z    ph, x, z

Proof of Theorem dvidlemap
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dvidlem.1 . . . . . 6  |-  ( ph  ->  F : CC --> CC )
2 cnex 7668 . . . . . . 7  |-  CC  e.  _V
32, 2fpm 6529 . . . . . 6  |-  ( F : CC --> CC  ->  F  e.  ( CC  ^pm  CC ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  CC ) )
5 dvfcnpm 12614 . . . . 5  |-  ( F  e.  ( CC  ^pm  CC )  ->  ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC )
64, 5syl 14 . . . 4  |-  ( ph  ->  ( CC  _D  F
) : dom  ( CC  _D  F ) --> CC )
7 ssidd 3084 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
87, 1, 7dvbss 12609 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
9 reldvg 12603 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  Rel  ( CC  _D  F
) )
107, 4, 9syl2anc 406 . . . . . . . 8  |-  ( ph  ->  Rel  ( CC  _D  F ) )
1110adantr 272 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  Rel  ( CC  _D  F ) )
12 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
13 eqid 2115 . . . . . . . . . . 11  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
1413cntoptop 12522 . . . . . . . . . 10  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  Top
1513cntoptopon 12521 . . . . . . . . . . . 12  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  (TopOn `  CC )
1615toponunii 12027 . . . . . . . . . . 11  |-  CC  =  U. ( MetOpen `  ( abs  o. 
-  ) )
1716ntrtop 12140 . . . . . . . . . 10  |-  ( (
MetOpen `  ( abs  o.  -  ) )  e. 
Top  ->  ( ( int `  ( MetOpen `  ( abs  o. 
-  ) ) ) `
 CC )  =  CC )
1814, 17ax-mp 7 . . . . . . . . 9  |-  ( ( int `  ( MetOpen `  ( abs  o.  -  )
) ) `  CC )  =  CC
1912, 18syl6eleqr 2208 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( MetOpen
`  ( abs  o.  -  ) ) ) `
 CC ) )
20 limcresi 12591 . . . . . . . . . 10  |-  ( ( z  e.  CC  |->  B ) lim CC  x ) 
C_  ( ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x } ) lim CC  x )
21 dvidlem.3 . . . . . . . . . . . 12  |-  B  e.  CC
22 ssidd 3084 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  CC  C_  CC )
23 cncfmptc 12568 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2421, 22, 22, 23mp3an2i 1303 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
25 eqidd 2116 . . . . . . . . . . 11  |-  ( z  =  x  ->  B  =  B )
2624, 12, 25cnmptlimc 12599 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2720, 26sseldi 3061 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
) lim CC  x )
)
28 breq1 3898 . . . . . . . . . . . . . 14  |-  ( w  =  z  ->  (
w #  x  <->  z #  x
) )
2928elrab 2809 . . . . . . . . . . . . 13  |-  ( z  e.  { w  e.  CC  |  w #  x } 
<->  ( z  e.  CC  /\  z #  x ) )
30 dvidlemap.2 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  CC  /\  z  e.  CC  /\  z #  x ) )  ->  (
( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) )  =  B )
31303exp2 1186 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z #  x  -> 
( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
3231imp43 350 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z #  x ) )  -> 
( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B )
3329, 32sylan2b 283 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  { w  e.  CC  |  w #  x }
)  ->  ( (
( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) )  =  B )
3433mpteq2dva 3978 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  { w  e.  CC  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )  =  ( z  e.  {
w  e.  CC  |  w #  x }  |->  B ) )
35 ssrab2 3148 . . . . . . . . . . . 12  |-  { w  e.  CC  |  w #  x }  C_  CC
36 resmpt 4825 . . . . . . . . . . . 12  |-  ( { w  e.  CC  |  w #  x }  C_  CC  ->  ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
)  =  ( z  e.  { w  e.  CC  |  w #  x }  |->  B ) )
3735, 36ax-mp 7 . . . . . . . . . . 11  |-  ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x } )  =  ( z  e.  { w  e.  CC  |  w #  x }  |->  B )
3834, 37syl6eqr 2165 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  { w  e.  CC  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )  =  ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
) )
3938oveq1d 5743 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( z  e.  { w  e.  CC  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) ) lim CC  x )  =  ( ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
) lim CC  x )
)
4027, 39eleqtrrd 2194 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e. 
{ w  e.  CC  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )
4115toponrestid 12031 . . . . . . . . 9  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( (
MetOpen `  ( abs  o.  -  ) )t  CC )
42 eqid 2115 . . . . . . . . 9  |-  ( z  e.  { w  e.  CC  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )  =  ( z  e.  {
w  e.  CC  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) )
431adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4441, 13, 42, 22, 43, 22eldvap 12606 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x ( CC  _D  F
) B  <->  ( x  e.  ( ( int `  ( MetOpen
`  ( abs  o.  -  ) ) ) `
 CC )  /\  B  e.  ( (
z  e.  { w  e.  CC  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) ) lim CC  x ) ) ) )
4519, 40, 44mpbir2and 911 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
46 releldm 4734 . . . . . . 7  |-  ( ( Rel  ( CC  _D  F )  /\  x
( CC  _D  F
) B )  ->  x  e.  dom  ( CC 
_D  F ) )
4711, 45, 46syl2anc 406 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e. 
dom  ( CC  _D  F ) )
488, 47eqelssd 3082 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4948feq2d 5218 . . . 4  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : CC --> CC ) )
506, 49mpbid 146 . . 3  |-  ( ph  ->  ( CC  _D  F
) : CC --> CC )
5150ffnd 5231 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5278 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5321, 52mp1i 10 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
546adantr 272 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC )
5554ffund 5234 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  Fun  ( CC  _D  F ) )
56 funbrfvb 5418 . . . . 5  |-  ( ( Fun  ( CC  _D  F )  /\  x  e.  dom  ( CC  _D  F ) )  -> 
( ( ( CC 
_D  F ) `  x )  =  B  <-> 
x ( CC  _D  F ) B ) )
5755, 47, 56syl2anc 406 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( ( CC  _D  F
) `  x )  =  B  <->  x ( CC 
_D  F ) B ) )
5845, 57mpbird 166 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  B )
5921a1i 9 . . . 4  |-  ( ph  ->  B  e.  CC )
60 fvconst2g 5588 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 279 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2150 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5475 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   {crab 2394    C_ wss 3037   {csn 3493   class class class wbr 3895    |-> cmpt 3949    X. cxp 4497   dom cdm 4499    |` cres 4501    o. ccom 4503   Rel wrel 4504   Fun wfun 5075    Fn wfn 5076   -->wf 5077   ` cfv 5081  (class class class)co 5728    ^pm cpm 6497   CCcc 7545    - cmin 7856   # cap 8261    / cdiv 8345   abscabs 10661   MetOpencmopn 11997   Topctop 12007   intcnt 12105   -cn->ccncf 12543   lim CC climc 12579    _D cdv 12580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-isom 5090  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-map 6498  df-pm 6499  df-sup 6823  df-inf 6824  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-xneg 9452  df-xadd 9453  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-rest 11965  df-topgen 11984  df-psmet 11999  df-xmet 12000  df-met 12001  df-bl 12002  df-mopn 12003  df-top 12008  df-topon 12021  df-bases 12053  df-ntr 12108  df-cn 12200  df-cnp 12201  df-cncf 12544  df-limced 12581  df-dvap 12582
This theorem is referenced by:  dvconst  12616  dvid  12617
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