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Theorem dvidlemap 14637
Description: Lemma for dvid 14639 and dvconst 14638. (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 7966 . . . . . . 7  |-  CC  e.  _V
32, 2fpm 6708 . . . . . 6  |-  ( F : CC --> CC  ->  F  e.  ( CC  ^pm  CC ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  CC ) )
5 dvfcnpm 14636 . . . . 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 3191 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
87, 1, 7dvbss 14631 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
9 reldvg 14625 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  Rel  ( CC  _D  F
) )
107, 4, 9syl2anc 411 . . . . . . . 8  |-  ( ph  ->  Rel  ( CC  _D  F ) )
1110adantr 276 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  Rel  ( CC  _D  F ) )
12 simpr 110 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
13 eqid 2189 . . . . . . . . . . 11  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
1413cntoptop 14510 . . . . . . . . . 10  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  Top
1513cntoptopon 14509 . . . . . . . . . . . 12  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  (TopOn `  CC )
1615toponunii 13994 . . . . . . . . . . 11  |-  CC  =  U. ( MetOpen `  ( abs  o. 
-  ) )
1716ntrtop 14105 . . . . . . . . . 10  |-  ( (
MetOpen `  ( abs  o.  -  ) )  e. 
Top  ->  ( ( int `  ( MetOpen `  ( abs  o. 
-  ) ) ) `
 CC )  =  CC )
1814, 17ax-mp 5 . . . . . . . . 9  |-  ( ( int `  ( MetOpen `  ( abs  o.  -  )
) ) `  CC )  =  CC
1912, 18eleqtrrdi 2283 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( MetOpen
`  ( abs  o.  -  ) ) ) `
 CC ) )
20 limcresi 14612 . . . . . . . . . 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 3191 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  CC  C_  CC )
23 cncfmptc 14559 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2421, 22, 22, 23mp3an2i 1353 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
25 eqidd 2190 . . . . . . . . . . 11  |-  ( z  =  x  ->  B  =  B )
2624, 12, 25cnmptlimc 14620 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2720, 26sselid 3168 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
) lim CC  x )
)
28 breq1 4021 . . . . . . . . . . . . . 14  |-  ( w  =  z  ->  (
w #  x  <->  z #  x
) )
2928elrab 2908 . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  CC  ->  ( z  e.  CC  ->  ( z #  x  -> 
( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B ) ) ) )
3231imp43 355 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  CC )  /\  (
z  e.  CC  /\  z #  x ) )  -> 
( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B )
3329, 32sylan2b 287 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  CC )  /\  z  e.  { w  e.  CC  |  w #  x }
)  ->  ( (
( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) )  =  B )
3433mpteq2dva 4108 . . . . . . . . . . 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 3255 . . . . . . . . . . . 12  |-  { w  e.  CC  |  w #  x }  C_  CC
36 resmpt 4973 . . . . . . . . . . . 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 5 . . . . . . . . . . 11  |-  ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x } )  =  ( z  e.  { w  e.  CC  |  w #  x }  |->  B )
3834, 37eqtr4di 2240 . . . . . . . . . 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 5912 . . . . . . . . 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 2269 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e. 
{ w  e.  CC  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )
4115toponrestid 13998 . . . . . . . . 9  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( (
MetOpen `  ( abs  o.  -  ) )t  CC )
42 eqid 2189 . . . . . . . . 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 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  F : CC
--> CC )
4441, 13, 42, 22, 43, 22eldvap 14628 . . . . . . . 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 946 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
46 releldm 4880 . . . . . . 7  |-  ( ( Rel  ( CC  _D  F )  /\  x
( CC  _D  F
) B )  ->  x  e.  dom  ( CC 
_D  F ) )
4711, 45, 46syl2anc 411 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e. 
dom  ( CC  _D  F ) )
488, 47eqelssd 3189 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4948feq2d 5372 . . . 4  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : CC --> CC ) )
506, 49mpbid 147 . . 3  |-  ( ph  ->  ( CC  _D  F
) : CC --> CC )
5150ffnd 5385 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5432 . . 3  |-  ( B  e.  CC  ->  ( CC  X.  { B }
)  Fn  CC )
5321, 52mp1i 10 . 2  |-  ( ph  ->  ( CC  X.  { B } )  Fn  CC )
546adantr 276 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC )
5554ffund 5388 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  Fun  ( CC  _D  F ) )
56 funbrfvb 5579 . . . . 5  |-  ( ( Fun  ( CC  _D  F )  /\  x  e.  dom  ( CC  _D  F ) )  -> 
( ( ( CC 
_D  F ) `  x )  =  B  <-> 
x ( CC  _D  F ) B ) )
5755, 47, 56syl2anc 411 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( ( CC  _D  F
) `  x )  =  B  <->  x ( CC 
_D  F ) B ) )
5845, 57mpbird 167 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  B )
5921a1i 9 . . . 4  |-  ( ph  ->  B  e.  CC )
60 fvconst2g 5751 . . . 4  |-  ( ( B  e.  CC  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x )  =  B )
6159, 60sylan 283 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  X.  { B } ) `  x
)  =  B )
6258, 61eqtr4d 2225 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5637 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   {crab 2472    C_ wss 3144   {csn 3607   class class class wbr 4018    |-> cmpt 4079    X. cxp 4642   dom cdm 4644    |` cres 4646    o. ccom 4648   Rel wrel 4649   Fun wfun 5229    Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5897    ^pm cpm 6676   CCcc 7840    - cmin 8159   # cap 8569    / cdiv 8660   abscabs 11041   MetOpencmopn 13871   Topctop 13974   intcnt 14070   -cn->ccncf 14534   lim CC climc 14600    _D cdv 14601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-map 6677  df-pm 6678  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-xneg 9804  df-xadd 9805  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-rest 12749  df-topgen 12768  df-psmet 13873  df-xmet 13874  df-met 13875  df-bl 13876  df-mopn 13877  df-top 13975  df-topon 13988  df-bases 14020  df-ntr 14073  df-cn 14165  df-cnp 14166  df-cncf 14535  df-limced 14602  df-dvap 14603
This theorem is referenced by:  dvconst  14638  dvid  14639
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