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

Theorem rlimeq 12059
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
Hypotheses
Ref Expression
rlimeq.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
rlimeq.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
rlimeq.3  |-  ( ph  ->  D  e.  RR )
rlimeq.4  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
Assertion
Ref Expression
rlimeq  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
Distinct variable groups:    x, A    x, D    ph, x
Allowed substitution hints:    B( x)    C( x)    E( x)

Proof of Theorem rlimeq
StepHypRef Expression
1 rlimss 11992 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  E  ->  dom  ( x  e.  A  |->  B )  C_  RR )
2 rlimeq.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3 eqid 2296 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 5700 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
5 fdm 5409 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> CC  ->  dom  ( x  e.  A  |->  B )  =  A )
64, 5syl 15 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
76sseq1d 3218 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
81, 7syl5ib 210 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  ->  A  C_  RR )
)
9 rlimss 11992 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  dom  ( x  e.  A  |->  C )  C_  RR )
10 rlimeq.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
11 eqid 2296 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 5700 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> CC )
13 fdm 5409 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A --> CC  ->  dom  ( x  e.  A  |->  C )  =  A )
1412, 13syl 15 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1514sseq1d 3218 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  C ) 
C_  RR  <->  A  C_  RR ) )
169, 15syl5ib 210 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  ~~> r  E  ->  A  C_  RR )
)
17 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,)  +oo ) ) )
18 elin 3371 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  i^i  ( D [,)  +oo )
)  <->  ( x  e.  A  /\  x  e.  ( D [,)  +oo ) ) )
1917, 18sylib 188 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  ( x  e.  A  /\  x  e.  ( D [,)  +oo ) ) )
2019simpld 445 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  A
)
2119simprd 449 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  ( D [,)  +oo )
)
22 rlimeq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
23 elicopnf 10755 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  (
x  e.  ( D [,)  +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2422, 23syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( x  e.  ( D [,)  +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2524biimpa 470 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,)  +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2621, 25syldan 456 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2726simprd 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  D  <_  x
)
2820, 27jca 518 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  ( x  e.  A  /\  D  <_  x ) )
29 rlimeq.4 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
3028, 29syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  B  =  C )
3130mpteq2dva 4122 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  B )  =  ( x  e.  ( A  i^i  ( D [,)  +oo )
)  |->  C ) )
32 inss1 3402 . . . . . . . . . 10  |-  ( A  i^i  ( D [,)  +oo ) )  C_  A
33 resmpt 5016 . . . . . . . . . 10  |-  ( ( A  i^i  ( D [,)  +oo ) )  C_  A  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  B ) )
3432, 33ax-mp 8 . . . . . . . . 9  |-  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  B )
35 resmpt 5016 . . . . . . . . . 10  |-  ( ( A  i^i  ( D [,)  +oo ) )  C_  A  ->  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  C ) )
3632, 35ax-mp 8 . . . . . . . . 9  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  C )
3731, 34, 363eqtr4g 2353 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) ) )
38 resres 4984 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )
39 resres 4984 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) )
4037, 38, 393eqtr4g 2353 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) ) )
41 ssid 3210 . . . . . . . 8  |-  A  C_  A
42 resmpt 5016 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 4965 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B )  -> 
( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo )
) )
4441, 42, 43mp2b 9 . . . . . . 7  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )
45 resmpt 5016 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 4965 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )  -> 
( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo )
) )
4741, 45, 46mp2b 9 . . . . . . 7  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )
4840, 44, 473eqtr3g 2351 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) ) )
4948breq1d 4049 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5049adantr 451 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
514adantr 451 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> CC )
52 simpr 447 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5322adantr 451 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5451, 52, 53rlimresb 12055 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( (
x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5512adantr 451 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> CC )
5655, 52, 53rlimresb 12055 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  C )  ~~> r  E  <->  ( (
x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5750, 54, 563bitr4d 276 . . 3  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
5857ex 423 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) ) )
598, 16, 58pm5.21ndd 343 1  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705    |` cres 4707   -->wf 5267  (class class class)co 5874   CCcc 8751   RRcr 8752    +oocpnf 8880    <_ cle 8884   [,)cico 10674    ~~> r crli 11975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ico 10678  df-rlim 11979
  Copyright terms: Public domain W3C validator