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Theorem o1lo12 12102
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 12094 . . 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 12083 . . 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 2702 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
7 dmmptg 5249 . . . . 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 3281 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
10 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
115renegcld 9297 . . . . . . 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 9297 . . . . . 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 9438 . . . . . . . 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 12087 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  -u B )  e. 
<_ O ( 1 ) )
225o1lo1 12101 . . . . . 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 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   class class class wbr 4102    e. cmpt 4156   dom cdm 4768   RRcr 8823    <_ cle 8955   -ucneg 9125   O (
1 )co1 12050   <_ O ( 1 )clo1 12051
This theorem is referenced by:  dirith2  20783  vmalogdivsum2  20793  pntrlog2bndlem4  20835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-ico 10751  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-o1 12054  df-lo1 12055
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