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Theorem lo1eq 13049
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 13000 . . 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 2443 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 5870 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
5 fdm 5566 . . . . 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 3386 . . 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 13000 . . 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 2443 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 5870 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> RR )
13 fdm 5566 . . . . 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 3386 . . 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 3542 . . . . . . . . . . . . . 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 11388 . . . . . . . . . . . . . . . 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 4381 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
32 inss1 3573 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
33 resmpt 5159 . . . . . . . . . 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 5159 . . . . . . . . . 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 2500 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
38 resres 5126 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
39 resres 5126 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
4037, 38, 393eqtr4g 2500 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
41 ssid 3378 . . . . . . . 8  |-  A  C_  A
42 resmpt 5159 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 5107 . . . . . . . 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 5159 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 5107 . . . . . . . 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 2498 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4948eleq1d 2509 . . . . 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 13045 . . . 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 13045 . . . 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 3330    C_ wss 3331   class class class wbr 4295    e. cmpt 4353   dom cdm 4843    |` cres 4845   -->wf 5417  (class class class)co 6094   RRcr 9284   +oocpnf 9418    <_ cle 9422   [,)cico 11305   <_O(1)clo1 12968
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 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-pre-lttri 9359  ax-pre-lttrn 9360
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-ico 11309  df-lo1 12972
This theorem is referenced by:  o1eq  13051
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