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Theorem lo1eq 13393
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 13344 . . 3  |-  ( ( x  e.  A  |->  B )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  B )  C_  RR )
2 eqid 2382 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
3 lo1eq.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
42, 3dmmptd 5619 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
54sseq1d 3444 . . 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 13344 . . 3  |-  ( ( x  e.  A  |->  C )  e.  <_O(1)  ->  dom  ( x  e.  A  |->  C )  C_  RR )
8 eqid 2382 . . . . 5  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
9 lo1eq.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
108, 9dmmptd 5619 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1110sseq1d 3444 . . 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 459 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,) +oo ) ) )
14 elin 3601 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  A )
1715simprd 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( D [,) +oo )
)
18 lo1eq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
19 elicopnf 11541 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,) +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2217, 21syldan 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2322simprd 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  D  <_  x )
2416, 23jca 530 . . . . . . . . . . 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 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  B  =  C )
2726mpteq2dva 4453 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
28 inss1 3632 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
29 resmpt 5235 . . . . . . . . . 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 5235 . . . . . . . . . 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 2448 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
34 resres 5198 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
35 resres 5198 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
3633, 34, 353eqtr4g 2448 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
37 ssid 3436 . . . . . . . 8  |-  A  C_  A
38 resmpt 5235 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
39 reseq1 5180 . . . . . . . 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 5235 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
42 reseq1 5180 . . . . . . . 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 2446 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4544eleq1d 2451 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1)  <->  (
( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
4645adantr 463 . . . 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 5957 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
4847adantr 463 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> RR )
49 simpr 459 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5018adantr 463 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5148, 49, 50lo1resb 13389 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  <_O(1)  <->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  e.  <_O(1) ) )
529, 8fmptd 5957 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> RR )
5352adantr 463 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> RR )
5453, 49, 50lo1resb 13389 . . . 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 432 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  e. 
<_O(1)  <-> 
( x  e.  A  |->  C )  e.  <_O(1) ) ) )
576, 12, 56pm5.21ndd 352 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 367    = wceq 1399    e. wcel 1826    i^i cin 3388    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   dom cdm 4913    |` cres 4915   -->wf 5492  (class class class)co 6196   RRcr 9402   +oocpnf 9536    <_ cle 9540   [,)cico 11452   <_O(1)clo1 13312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-pre-lttri 9477  ax-pre-lttrn 9478
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-ico 11456  df-lo1 13316
This theorem is referenced by:  o1eq  13395
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