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Theorem lo1eq 13371
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 13322 . . 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 2467 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 6056 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
5 fdm 5741 . . . . 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 3536 . . 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 13322 . . 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 2467 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 6056 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> RR )
13 fdm 5741 . . . . 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 3536 . . 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 3692 . . . . . . . . . . . . . 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 11632 . . . . . . . . . . . . . . . 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 4539 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
32 inss1 3723 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
33 resmpt 5329 . . . . . . . . . 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 5329 . . . . . . . . . 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 2533 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
38 resres 5292 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
39 resres 5292 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
4037, 38, 393eqtr4g 2533 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
41 ssid 3528 . . . . . . . 8  |-  A  C_  A
42 resmpt 5329 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 5273 . . . . . . . 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 5329 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 5273 . . . . . . . 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 2531 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4948eleq1d 2536 . . . . 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 13367 . . . 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 13367 . . . 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 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   dom cdm 5005    |` cres 5007   -->wf 5590  (class class class)co 6295   RRcr 9503   +oocpnf 9637    <_ cle 9641   [,)cico 11543   <_O(1)clo1 13290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-ico 11547  df-lo1 13294
This theorem is referenced by:  o1eq  13373
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