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Theorem lo1sub 12455
Description: The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply  ( x  e.  A  |->  -u C
)  e.  <_ O
( 1 ), so it is just a special case of lo1add 12451. (Contributed by Mario Carneiro, 31-May-2016.)
Hypotheses
Ref Expression
lo1sub.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
lo1sub.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
lo1sub.3  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
lo1sub.4  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O
( 1 ) )
Assertion
Ref Expression
lo1sub  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  e.  <_ O ( 1 ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem lo1sub
StepHypRef Expression
1 lo1sub.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
2 lo1sub.3 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
31, 2lo1mptrcl 12446 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
43recnd 9145 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
5 lo1sub.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  RR )
65recnd 9145 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
74, 6negsubd 9448 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( B  +  -u C )  =  ( B  -  C ) )
87mpteq2dva 4320 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( B  +  -u C ) )  =  ( x  e.  A  |->  ( B  -  C
) ) )
95renegcld 9495 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  -u C  e.  RR )
10 lo1sub.4 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O
( 1 ) )
115o1lo1 12362 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e.  O ( 1 )  <-> 
( ( x  e.  A  |->  C )  e. 
<_ O ( 1 )  /\  ( x  e.  A  |->  -u C )  e. 
<_ O ( 1 ) ) ) )
1210, 11mpbid 203 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e. 
<_ O ( 1 )  /\  ( x  e.  A  |->  -u C )  e. 
<_ O ( 1 ) ) )
1312simprd 451 . . 3  |-  ( ph  ->  ( x  e.  A  |-> 
-u C )  e. 
<_ O ( 1 ) )
143, 9, 2, 13lo1add 12451 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( B  +  -u C ) )  e. 
<_ O ( 1 ) )
158, 14eqeltrrd 2517 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  e.  <_ O ( 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727    e. cmpt 4291  (class class class)co 6110   RRcr 9020    + caddc 9024    - cmin 9322   -ucneg 9323   O ( 1 )co1 12311   <_ O ( 1 )clo1 12312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-pm 7050  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-ico 10953  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-o1 12315  df-lo1 12316
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