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Theorem dvidlemap 13731
Description: Lemma for dvid 13733 and dvconst 13732. (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 7910 . . . . . . 7  |-  CC  e.  _V
32, 2fpm 6671 . . . . . 6  |-  ( F : CC --> CC  ->  F  e.  ( CC  ^pm  CC ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  CC ) )
5 dvfcnpm 13730 . . . . 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 3174 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
87, 1, 7dvbss 13725 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  C_  CC )
9 reldvg 13719 . . . . . . . . 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 2175 . . . . . . . . . . 11  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
1413cntoptop 13604 . . . . . . . . . 10  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  Top
1513cntoptopon 13603 . . . . . . . . . . . 12  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  (TopOn `  CC )
1615toponunii 13086 . . . . . . . . . . 11  |-  CC  =  U. ( MetOpen `  ( abs  o. 
-  ) )
1716ntrtop 13199 . . . . . . . . . 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 2269 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  ( ( int `  ( MetOpen
`  ( abs  o.  -  ) ) ) `
 CC ) )
20 limcresi 13706 . . . . . . . . . 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 3174 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  CC )  ->  CC  C_  CC )
23 cncfmptc 13653 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
z  e.  CC  |->  B )  e.  ( CC
-cn-> CC ) )
2421, 22, 22, 23mp3an2i 1342 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( z  e.  CC  |->  B )  e.  ( CC -cn-> CC ) )
25 eqidd 2176 . . . . . . . . . . 11  |-  ( z  =  x  ->  B  =  B )
2624, 12, 25cnmptlimc 13714 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e.  CC  |->  B ) lim CC  x ) )
2720, 26sselid 3151 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( ( z  e.  CC  |->  B )  |`  { w  e.  CC  |  w #  x }
) lim CC  x )
)
28 breq1 4001 . . . . . . . . . . . . . 14  |-  ( w  =  z  ->  (
w #  x  <->  z #  x
) )
2928elrab 2891 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . 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 4088 . . . . . . . . . . 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 3238 . . . . . . . . . . . 12  |-  { w  e.  CC  |  w #  x }  C_  CC
36 resmpt 4948 . . . . . . . . . . . 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 2226 . . . . . . . . . 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 5880 . . . . . . . . 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 2255 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  ( ( z  e. 
{ w  e.  CC  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )
4115toponrestid 13090 . . . . . . . . 9  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( (
MetOpen `  ( abs  o.  -  ) )t  CC )
42 eqid 2175 . . . . . . . . 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 13722 . . . . . . . 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 944 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  x ( CC  _D  F ) B )
46 releldm 4855 . . . . . . 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 3172 . . . . 5  |-  ( ph  ->  dom  ( CC  _D  F )  =  CC )
4948feq2d 5345 . . . 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 5358 . 2  |-  ( ph  ->  ( CC  _D  F
)  Fn  CC )
52 fnconstg 5405 . . 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 5361 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  Fun  ( CC  _D  F ) )
56 funbrfvb 5550 . . . . 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 5722 . . . 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 2211 . 2  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( CC  _D  F ) `
 x )  =  ( ( CC  X.  { B } ) `  x ) )
6351, 53, 62eqfnfvd 5608 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
X.  { B }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   {crab 2457    C_ wss 3127   {csn 3589   class class class wbr 3998    |-> cmpt 4059    X. cxp 4618   dom cdm 4620    |` cres 4622    o. ccom 4624   Rel wrel 4625   Fun wfun 5202    Fn wfn 5203   -->wf 5204   ` cfv 5208  (class class class)co 5865    ^pm cpm 6639   CCcc 7784    - cmin 8102   # cap 8512    / cdiv 8602   abscabs 10974   MetOpencmopn 13056   Topctop 13066   intcnt 13164   -cn->ccncf 13628   lim CC climc 13694    _D cdv 13695
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-pm 6641  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-xneg 9743  df-xadd 9744  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-rest 12612  df-topgen 12631  df-psmet 13058  df-xmet 13059  df-met 13060  df-bl 13061  df-mopn 13062  df-top 13067  df-topon 13080  df-bases 13112  df-ntr 13167  df-cn 13259  df-cnp 13260  df-cncf 13629  df-limced 13696  df-dvap 13697
This theorem is referenced by:  dvconst  13732  dvid  13733
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