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

Theorem lo1eq 13365
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 13316 . . 3  |-  ( ( x  e.  A  |->  B )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  B )  C_  RR )
2 eqid 2441 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
3 lo1eq.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
42, 3dmmptd 5697 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
54sseq1d 3513 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
61, 5syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  ->  A  C_  RR ) )
7 lo1dm 13316 . . 3  |-  ( ( x  e.  A  |->  C )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  C )  C_  RR )
8 eqid 2441 . . . . 5  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
9 lo1eq.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
108, 9dmmptd 5697 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1110sseq1d 3513 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  C ) 
C_  RR  <->  A  C_  RR ) )
127, 11syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e. 
<_O(1)  ->  A  C_  RR ) )
13 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,) +oo ) ) )
14 elin 3669 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  i^i  ( D [,) +oo )
)  <->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
1513, 14sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
1615simpld 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  A )
1715simprd 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( D [,) +oo )
)
18 lo1eq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
19 elicopnf 11624 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  (
x  e.  ( D [,) +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2018, 19syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( x  e.  ( D [,) +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2120biimpa 484 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,) +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2217, 21syldan 470 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2322simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  D  <_  x )
2416, 23jca 532 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  D  <_  x ) )
25 lo1eq.4 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
2624, 25syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  B  =  C )
2726mpteq2dva 4519 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
28 inss1 3700 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
29 resmpt 5309 . . . . . . . . . 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 ) )
3028, 29ax-mp 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )
31 resmpt 5309 . . . . . . . . . 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 ) )
3228, 31ax-mp 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  C )
3327, 30, 323eqtr4g 2507 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
34 resres 5272 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
35 resres 5272 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
3633, 34, 353eqtr4g 2507 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
37 ssid 3505 . . . . . . . 8  |-  A  C_  A
38 resmpt 5309 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
39 reseq1 5253 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B )  -> 
( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) ) )
4037, 38, 39mp2b 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )
41 resmpt 5309 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
42 reseq1 5253 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )  -> 
( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4337, 41, 42mp2b 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )
4436, 40, 433eqtr3g 2505 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4544eleq1d 2510 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1)  <->  (
( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
4645adantr 465 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e. 
<_O(1)  <-> 
( ( x  e.  A  |->  C )  |`  ( D [,) +oo )
)  e.  <_O(1) ) )
473, 2fmptd 6036 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
4847adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> RR )
49 simpr 461 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5018adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5148, 49, 50lo1resb 13361 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  <_O(1)  <->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
529, 8fmptd 6036 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> RR )
5352adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> RR )
5453, 49, 50lo1resb 13361 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  C )  e.  <_O(1)  <->  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
5546, 51, 543bitr4d 285 . . 3  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  <_O(1)  <->  ( x  e.  A  |->  C )  e. 
<_O(1) ) )
5655ex 434 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  <-> 
( x  e.  A  |->  C )  e.  <_O(1) ) ) )
576, 12, 56pm5.21ndd 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 1381    e. wcel 1802    i^i cin 3457    C_ wss 3458   class class class wbr 4433    |-> cmpt 4491   dom cdm 4985    |` cres 4987   -->wf 5570  (class class class)co 6277   RRcr 9489   +oocpnf 9623    <_ cle 9627   [,)cico 11535   <_O(1)clo1 13284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-pre-lttri 9564  ax-pre-lttrn 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-ico 11539  df-lo1 13288
This theorem is referenced by:  o1eq  13367
  Copyright terms: Public domain W3C validator