Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reclimc Structured version   Visualization version   Unicode version

Theorem reclimc 37772
Description: Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
reclimc.f  |-  F  =  ( x  e.  A  |->  B )
reclimc.g  |-  G  =  ( x  e.  A  |->  ( 1  /  B
) )
reclimc.b  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  ( CC  \  {
0 } ) )
reclimc.c  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
reclimc.cne0  |-  ( ph  ->  C  =/=  0 )
Assertion
Ref Expression
reclimc  |-  ( ph  ->  ( 1  /  C
)  e.  ( G lim
CC  D ) )
Distinct variable groups:    x, A    x, C    x, D    ph, x
Allowed substitution hints:    B( x)    F( x)    G( x)

Proof of Theorem reclimc
StepHypRef Expression
1 eqid 2462 . . . 4  |-  ( x  e.  A  |->  ( C  -  B ) )  =  ( x  e.  A  |->  ( C  -  B ) )
2 eqid 2462 . . . 4  |-  ( x  e.  A  |->  ( B  x.  C ) )  =  ( x  e.  A  |->  ( B  x.  C ) )
3 eqid 2462 . . . 4  |-  ( x  e.  A  |->  ( ( C  -  B )  /  ( B  x.  C ) ) )  =  ( x  e.  A  |->  ( ( C  -  B )  / 
( B  x.  C
) ) )
4 limccl 22879 . . . . . . 7  |-  ( F lim
CC  D )  C_  CC
5 reclimc.c . . . . . . 7  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
64, 5sseldi 3442 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76adantr 471 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
8 reclimc.b . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  ( CC  \  {
0 } ) )
98eldifad 3428 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
107, 9subcld 10012 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( C  -  B )  e.  CC )
119, 7mulcld 9689 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  x.  C )  e.  CC )
12 eldifsni 4111 . . . . . . . . 9  |-  ( B  e.  ( CC  \  { 0 } )  ->  B  =/=  0
)
138, 12syl 17 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  B  =/=  0 )
14 reclimc.cne0 . . . . . . . . 9  |-  ( ph  ->  C  =/=  0 )
1514adantr 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  C  =/=  0 )
169, 7, 13, 15mulne0d 10292 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( B  x.  C )  =/=  0 )
1716neneqd 2640 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  -.  ( B  x.  C
)  =  0 )
18 elsncg 4003 . . . . . . 7  |-  ( ( B  x.  C )  e.  CC  ->  (
( B  x.  C
)  e.  { 0 }  <->  ( B  x.  C )  =  0 ) )
1911, 18syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  x.  C
)  e.  { 0 }  <->  ( B  x.  C )  =  0 ) )
2017, 19mtbird 307 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  -.  ( B  x.  C
)  e.  { 0 } )
2111, 20eldifd 3427 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( B  x.  C )  e.  ( CC  \  {
0 } ) )
22 eqid 2462 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
23 eqid 2462 . . . . . 6  |-  ( x  e.  A  |->  -u B
)  =  ( x  e.  A  |->  -u B
)
24 eqid 2462 . . . . . 6  |-  ( x  e.  A  |->  ( C  +  -u B ) )  =  ( x  e.  A  |->  ( C  +  -u B ) )
259negcld 9999 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  -u B  e.  CC )
26 reclimc.f . . . . . . . 8  |-  F  =  ( x  e.  A  |->  B )
2726, 9, 5limcmptdm 37753 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
28 limcrcl 22878 . . . . . . . . 9  |-  ( C  e.  ( F lim CC  D )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  D  e.  CC ) )
295, 28syl 17 . . . . . . . 8  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  D  e.  CC ) )
3029simp3d 1028 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3122, 27, 6, 30constlimc 37742 . . . . . 6  |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  C ) lim CC  D ) )
3226, 23, 9, 5neglimc 37766 . . . . . 6  |-  ( ph  -> 
-u C  e.  ( ( x  e.  A  |-> 
-u B ) lim CC  D ) )
3322, 23, 24, 7, 25, 31, 32addlimc 37767 . . . . 5  |-  ( ph  ->  ( C  +  -u C )  e.  ( ( x  e.  A  |->  ( C  +  -u B ) ) lim CC  D ) )
346negidd 10002 . . . . 5  |-  ( ph  ->  ( C  +  -u C )  =  0 )
357, 9negsubd 10018 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( C  +  -u B )  =  ( C  -  B ) )
3635mpteq2dva 4503 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( C  +  -u B ) )  =  ( x  e.  A  |->  ( C  -  B
) ) )
3736oveq1d 6330 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( C  +  -u B ) ) lim CC  D )  =  ( ( x  e.  A  |->  ( C  -  B
) ) lim CC  D
) )
3833, 34, 373eltr3d 2554 . . . 4  |-  ( ph  ->  0  e.  ( ( x  e.  A  |->  ( C  -  B ) ) lim CC  D ) )
3926, 22, 2, 9, 7, 5, 31mullimc 37734 . . . 4  |-  ( ph  ->  ( C  x.  C
)  e.  ( ( x  e.  A  |->  ( B  x.  C ) ) lim CC  D ) )
406, 6, 14, 14mulne0d 10292 . . . 4  |-  ( ph  ->  ( C  x.  C
)  =/=  0 )
411, 2, 3, 10, 21, 38, 39, 400ellimcdiv 37768 . . 3  |-  ( ph  ->  0  e.  ( ( x  e.  A  |->  ( ( C  -  B
)  /  ( B  x.  C ) ) ) lim CC  D ) )
42 1cnd 9685 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  1  e.  CC )
4342, 9, 42, 7, 13, 15divsubdivd 10456 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( 1  /  B
)  -  ( 1  /  C ) )  =  ( ( ( 1  x.  C )  -  ( 1  x.  B ) )  / 
( B  x.  C
) ) )
447mulid2d 9687 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
1  x.  C )  =  C )
459mulid2d 9687 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
1  x.  B )  =  B )
4644, 45oveq12d 6333 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( 1  x.  C
)  -  ( 1  x.  B ) )  =  ( C  -  B ) )
4746oveq1d 6330 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( 1  x.  C )  -  (
1  x.  B ) )  /  ( B  x.  C ) )  =  ( ( C  -  B )  / 
( B  x.  C
) ) )
4843, 47eqtr2d 2497 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( C  -  B
)  /  ( B  x.  C ) )  =  ( ( 1  /  B )  -  ( 1  /  C
) ) )
4948mpteq2dva 4503 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  ( ( C  -  B )  /  ( B  x.  C )
) )  =  ( x  e.  A  |->  ( ( 1  /  B
)  -  ( 1  /  C ) ) ) )
5049oveq1d 6330 . . 3  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( C  -  B )  / 
( B  x.  C
) ) ) lim CC  D )  =  ( ( x  e.  A  |->  ( ( 1  /  B )  -  (
1  /  C ) ) ) lim CC  D
) )
5141, 50eleqtrd 2542 . 2  |-  ( ph  ->  0  e.  ( ( x  e.  A  |->  ( ( 1  /  B
)  -  ( 1  /  C ) ) ) lim CC  D ) )
52 reclimc.g . . 3  |-  G  =  ( x  e.  A  |->  ( 1  /  B
) )
53 eqid 2462 . . 3  |-  ( x  e.  A  |->  ( ( 1  /  B )  -  ( 1  /  C ) ) )  =  ( x  e.  A  |->  ( ( 1  /  B )  -  ( 1  /  C
) ) )
549, 13reccld 10404 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
1  /  B )  e.  CC )
556, 14reccld 10404 . . 3  |-  ( ph  ->  ( 1  /  C
)  e.  CC )
5652, 53, 27, 54, 30, 55ellimcabssub0 37735 . 2  |-  ( ph  ->  ( ( 1  /  C )  e.  ( G lim CC  D )  <->  0  e.  ( ( x  e.  A  |->  ( ( 1  /  B
)  -  ( 1  /  C ) ) ) lim CC  D ) ) )
5751, 56mpbird 240 1  |-  ( ph  ->  ( 1  /  C
)  e.  ( G lim
CC  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    \ cdif 3413    C_ wss 3416   {csn 3980    |-> cmpt 4475   dom cdm 4853   -->wf 5597  (class class class)co 6315   CCcc 9563   0cc0 9565   1c1 9566    + caddc 9568    x. cmul 9570    - cmin 9886   -ucneg 9887    / cdiv 10297   lim CC climc 22866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fi 7951  df-sup 7982  df-inf 7983  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-fz 11814  df-seq 12246  df-exp 12305  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-plusg 15252  df-mulr 15253  df-starv 15254  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-rest 15370  df-topn 15371  df-topgen 15391  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-cnfld 19020  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cnp 20293  df-xms 21384  df-ms 21385  df-limc 22870
This theorem is referenced by:  divlimc  37775  fourierdlem62  38070
  Copyright terms: Public domain W3C validator