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Theorem o1sub 12409
Description: The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
Assertion
Ref Expression
o1sub  |-  ( ( F  e.  O ( 1 )  /\  G  e.  O ( 1 ) )  ->  ( F  o F  -  G
)  e.  O ( 1 ) )

Proof of Theorem o1sub
Dummy variables  x  y  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 readdcl 9073 . 2  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2 subcl 9305 . 2  |-  ( ( m  e.  CC  /\  n  e.  CC )  ->  ( m  -  n
)  e.  CC )
3 simp2l 983 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  m  e.  CC )
4 simp2r 984 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  n  e.  CC )
53, 4subcld 9411 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( m  -  n )  e.  CC )
65abscld 12238 . . . 4  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  (
m  -  n ) )  e.  RR )
73abscld 12238 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  m
)  e.  RR )
84abscld 12238 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  n
)  e.  RR )
97, 8readdcld 9115 . . . 4  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( ( abs `  m )  +  ( abs `  n ) )  e.  RR )
10 simp1l 981 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  x  e.  RR )
11 simp1r 982 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  y  e.  RR )
1210, 11readdcld 9115 . . . 4  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( x  +  y )  e.  RR )
133, 4abs2dif2d 12260 . . . 4  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  (
m  -  n ) )  <_  ( ( abs `  m )  +  ( abs `  n
) ) )
14 simp3l 985 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  m
)  <_  x )
15 simp3r 986 . . . . 5  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  n
)  <_  y )
167, 8, 10, 11, 14, 15le2addd 9644 . . . 4  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( ( abs `  m )  +  ( abs `  n ) )  <_  ( x  +  y ) )
176, 9, 12, 13, 16letrd 9227 . . 3  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC )  /\  (
( abs `  m
)  <_  x  /\  ( abs `  n )  <_  y ) )  ->  ( abs `  (
m  -  n ) )  <_  ( x  +  y ) )
18173expia 1155 . 2  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( m  e.  CC  /\  n  e.  CC ) )  -> 
( ( ( abs `  m )  <_  x  /\  ( abs `  n
)  <_  y )  ->  ( abs `  (
m  -  n ) )  <_  ( x  +  y ) ) )
191, 2, 18o1of2 12406 1  |-  ( ( F  e.  O ( 1 )  /\  G  e.  O ( 1 ) )  ->  ( F  o F  -  G
)  e.  O ( 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   RRcr 8989    + caddc 8993    <_ cle 9121    - cmin 9291   abscabs 12039   O ( 1 )co1 12280
This theorem is referenced by:  o1sub2  12419  o1dif  12423  vmadivsum  21176  rpvmasumlem  21181  selberglem1  21239  selberg2  21245  pntrsumo1  21259  selbergr  21262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-o1 12284
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