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Theorem rlimeq 12322
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 12255 . . 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 2408 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 5856 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
5 fdm 5558 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> CC  ->  dom  ( x  e.  A  |->  B )  =  A )
64, 5syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
76sseq1d 3339 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
81, 7syl5ib 211 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  ->  A  C_  RR )
)
9 rlimss 12255 . . 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 2408 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 5856 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> CC )
13 fdm 5558 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A --> CC  ->  dom  ( x  e.  A  |->  C )  =  A )
1412, 13syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1514sseq1d 3339 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  C ) 
C_  RR  <->  A  C_  RR ) )
169, 15syl5ib 211 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  ~~> r  E  ->  A  C_  RR )
)
17 simpr 448 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,)  +oo ) ) )
18 elin 3494 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  i^i  ( D [,)  +oo )
)  <->  ( x  e.  A  /\  x  e.  ( D [,)  +oo ) ) )
1917, 18sylib 189 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  ( x  e.  A  /\  x  e.  ( D [,)  +oo ) ) )
2019simpld 446 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  A
)
2119simprd 450 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  x  e.  ( D [,)  +oo )
)
22 rlimeq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
23 elicopnf 10960 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  (
x  e.  ( D [,)  +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2422, 23syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( x  e.  ( D [,)  +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2524biimpa 471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,)  +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2621, 25syldan 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2726simprd 450 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  D  <_  x
)
2820, 27jca 519 . . . . . . . . . . 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 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,)  +oo ) ) )  ->  B  =  C )
3130mpteq2dva 4259 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,)  +oo ) )  |->  B )  =  ( x  e.  ( A  i^i  ( D [,)  +oo )
)  |->  C ) )
32 inss1 3525 . . . . . . . . . 10  |-  ( A  i^i  ( D [,)  +oo ) )  C_  A
33 resmpt 5154 . . . . . . . . . 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 5154 . . . . . . . . . 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 2465 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) ) )
38 resres 5122 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,)  +oo ) ) )
39 resres 5122 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,)  +oo ) ) )
4037, 38, 393eqtr4g 2465 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) ) )
41 ssid 3331 . . . . . . . 8  |-  A  C_  A
42 resmpt 5154 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 5103 . . . . . . . 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 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )
45 resmpt 5154 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 5103 . . . . . . . 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 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,)  +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )
4840, 44, 473eqtr3g 2463 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) ) )
4948breq1d 4186 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5049adantr 452 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
514adantr 452 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> CC )
52 simpr 448 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5322adantr 452 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5451, 52, 53rlimresb 12318 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( (
x  e.  A  |->  B )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5512adantr 452 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> CC )
5655, 52, 53rlimresb 12318 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  C )  ~~> r  E  <->  ( (
x  e.  A  |->  C )  |`  ( D [,)  +oo ) )  ~~> r  E
) )
5750, 54, 563bitr4d 277 . . 3  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
5857ex 424 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) ) )
598, 16, 58pm5.21ndd 344 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   class class class wbr 4176    e. cmpt 4230   dom cdm 4841    |` cres 4843   -->wf 5413  (class class class)co 6044   CCcc 8948   RRcr 8949    +oocpnf 9077    <_ cle 9081   [,)cico 10878    ~~> r crli 12238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-pre-lttri 9024  ax-pre-lttrn 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-ico 10882  df-rlim 12242
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