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Theorem o1lo12 12012
Description: A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
o1lo1.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
o1lo12.2  |-  ( ph  ->  M  e.  RR )
o1lo12.3  |-  ( (
ph  /\  x  e.  A )  ->  M  <_  B )
Assertion
Ref Expression
o1lo12  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <-> 
( x  e.  A  |->  B )  e.  <_ O ( 1 ) ) )
Distinct variable groups:    x, A    x, M    ph, x
Allowed substitution hint:    B( x)

Proof of Theorem o1lo12
StepHypRef Expression
1 o1dm 12004 . . 3  |-  ( ( x  e.  A  |->  B )  e.  O ( 1 )  ->  dom  ( x  e.  A  |->  B )  C_  RR )
21a1i 10 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  ->  dom  ( x  e.  A  |->  B ) 
C_  RR ) )
3 lo1dm 11993 . . 3  |-  ( ( x  e.  A  |->  B )  e.  <_ O
( 1 )  ->  dom  ( x  e.  A  |->  B )  C_  RR )
43a1i 10 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e. 
<_ O ( 1 )  ->  dom  ( x  e.  A  |->  B ) 
C_  RR ) )
5 o1lo1.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
65ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
7 dmmptg 5170 . . . . 5  |-  ( A. x  e.  A  B  e.  RR  ->  dom  ( x  e.  A  |->  B )  =  A )
86, 7syl 15 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
98sseq1d 3205 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
10 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
115renegcld 9210 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  -u B  e.  RR )
1211adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  A  C_  RR )  /\  x  e.  A )  ->  -u B  e.  RR )
13 o1lo12.2 . . . . . . 7  |-  ( ph  ->  M  e.  RR )
1413adantr 451 . . . . . 6  |-  ( (
ph  /\  A  C_  RR )  ->  M  e.  RR )
1514renegcld 9210 . . . . . 6  |-  ( (
ph  /\  A  C_  RR )  ->  -u M  e.  RR )
16 o1lo12.3 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  M  <_  B )
1713adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  M  e.  RR )
1817, 5lenegd 9351 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( M  <_  B  <->  -u B  <_  -u M ) )
1916, 18mpbid 201 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  -u B  <_ 
-u M )
2019ad2ant2r 727 . . . . . 6  |-  ( ( ( ph  /\  A  C_  RR )  /\  (
x  e.  A  /\  M  <_  x ) )  ->  -u B  <_  -u M
)
2110, 12, 14, 15, 20ello1d 11997 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  -u B )  e. 
<_ O ( 1 ) )
225o1lo1 12011 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <-> 
( ( x  e.  A  |->  B )  e. 
<_ O ( 1 )  /\  ( x  e.  A  |->  -u B )  e. 
<_ O ( 1 ) ) ) )
2322rbaibd 876 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  |->  -u B
)  e.  <_ O
( 1 ) )  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  B )  e. 
<_ O ( 1 ) ) )
2421, 23syldan 456 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  B )  e. 
<_ O ( 1 ) ) )
2524ex 423 . . 3  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <-> 
( x  e.  A  |->  B )  e.  <_ O ( 1 ) ) ) )
269, 25sylbid 206 . 2  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) ) ) )
272, 4, 26pm5.21ndd 343 1  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <-> 
( x  e.  A  |->  B )  e.  <_ O ( 1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   RRcr 8736    <_ cle 8868   -ucneg 9038   O (
1 )co1 11960   <_ O ( 1 )clo1 11961
This theorem is referenced by:  dirith2  20677  vmalogdivsum2  20687  pntrlog2bndlem4  20729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-o1 11964  df-lo1 11965
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