MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lo1eq Structured version   Unicode version

Theorem lo1eq 13038
Description: Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
lo1eq.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
lo1eq.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
lo1eq.3  |-  ( ph  ->  D  e.  RR )
lo1eq.4  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
Assertion
Ref Expression
lo1eq  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  <-> 
( x  e.  A  |->  C )  e.  <_O(1) ) )
Distinct variable groups:    x, A    x, D    ph, x
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem lo1eq
StepHypRef Expression
1 lo1dm 12989 . . 3  |-  ( ( x  e.  A  |->  B )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  B )  C_  RR )
2 lo1eq.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
3 eqid 2437 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 5860 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
5 fdm 5556 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> RR  ->  dom  ( x  e.  A  |->  B )  =  A )
64, 5syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
76sseq1d 3376 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
81, 7syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  ->  A  C_  RR ) )
9 lo1dm 12989 . . 3  |-  ( ( x  e.  A  |->  C )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  C )  C_  RR )
10 lo1eq.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
11 eqid 2437 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 5860 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> RR )
13 fdm 5556 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A --> RR  ->  dom  ( x  e.  A  |->  C )  =  A )
1412, 13syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1514sseq1d 3376 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  C ) 
C_  RR  <->  A  C_  RR ) )
169, 15syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e. 
<_O(1)  ->  A  C_  RR ) )
17 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,) +oo ) ) )
18 elin 3532 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  i^i  ( D [,) +oo )
)  <->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
1917, 18sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
2019simpld 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  A )
2119simprd 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( D [,) +oo )
)
22 lo1eq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
23 elicopnf 11377 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,) +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2621, 25syldan 470 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2726simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  D  <_  x )
2820, 27jca 532 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  D  <_  x ) )
29 lo1eq.4 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
3028, 29syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  B  =  C )
3130mpteq2dva 4371 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
32 inss1 3563 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
33 resmpt 5149 . . . . . . . . . 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 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )
35 resmpt 5149 . . . . . . . . . 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 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  C )
3731, 34, 363eqtr4g 2494 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
38 resres 5116 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
39 resres 5116 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
4037, 38, 393eqtr4g 2494 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
41 ssid 3368 . . . . . . . 8  |-  A  C_  A
42 resmpt 5149 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 5096 . . . . . . . 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 5149 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 5096 . . . . . . . 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 2492 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4948eleq1d 2503 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1)  <->  (
( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
5049adantr 465 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e. 
<_O(1)  <-> 
( ( x  e.  A  |->  C )  |`  ( D [,) +oo )
)  e.  <_O(1) ) )
514adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> RR )
52 simpr 461 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5322adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5451, 52, 53lo1resb 13034 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  <_O(1)  <->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
5512adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> RR )
5655, 52, 53lo1resb 13034 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  C )  e.  <_O(1)  <->  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
5750, 54, 563bitr4d 285 . . 3  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  <_O(1)  <->  ( x  e.  A  |->  C )  e. 
<_O(1) ) )
5857ex 434 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  <-> 
( x  e.  A  |->  C )  e.  <_O(1) ) ) )
598, 16, 58pm5.21ndd 354 1  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  <-> 
( x  e.  A  |->  C )  e.  <_O(1) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3320    C_ wss 3321   class class class wbr 4285    e. cmpt 4343   dom cdm 4832    |` cres 4834   -->wf 5407  (class class class)co 6086   RRcr 9273   +oocpnf 9407    <_ cle 9411   [,)cico 11294   <_O(1)clo1 12957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-pre-lttri 9348  ax-pre-lttrn 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-po 4633  df-so 4634  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-ico 11298  df-lo1 12961
This theorem is referenced by:  o1eq  13040
  Copyright terms: Public domain W3C validator