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Theorem o1dif 13128
Description: If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
o1dif.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
o1dif.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
o1dif.3  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  e.  O(1) )
Assertion
Ref Expression
o1dif  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O(1)  <-> 
( x  e.  A  |->  C )  e.  O(1) ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem o1dif
StepHypRef Expression
1 o1dif.3 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  e.  O(1) )
2 o1sub 13114 . . . . 5  |-  ( ( ( x  e.  A  |->  B )  e.  O(1)  /\  ( x  e.  A  |->  ( B  -  C
) )  e.  O(1) )  ->  ( ( x  e.  A  |->  B )  oF  -  (
x  e.  A  |->  ( B  -  C ) ) )  e.  O(1) )
32expcom 435 . . . 4  |-  ( ( x  e.  A  |->  ( B  -  C ) )  e.  O(1)  ->  (
( x  e.  A  |->  B )  e.  O(1)  -> 
( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  e.  O(1) ) )
41, 3syl 16 . . 3  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O(1)  ->  ( ( x  e.  A  |->  B )  oF  -  (
x  e.  A  |->  ( B  -  C ) ) )  e.  O(1) ) )
5 o1dif.1 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
6 o1dif.2 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
75, 6subcld 9740 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
87ralrimiva 2820 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  ( B  -  C
)  e.  CC )
9 dmmptg 5356 . . . . . . . . 9  |-  ( A. x  e.  A  ( B  -  C )  e.  CC  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  =  A )
108, 9syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  =  A )
11 o1dm 13029 . . . . . . . . 9  |-  ( ( x  e.  A  |->  ( B  -  C ) )  e.  O(1)  ->  dom  ( x  e.  A  |->  ( B  -  C
) )  C_  RR )
121, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  C_  RR )
1310, 12eqsstr3d 3412 . . . . . . 7  |-  ( ph  ->  A  C_  RR )
14 reex 9394 . . . . . . . 8  |-  RR  e.  _V
1514ssex 4457 . . . . . . 7  |-  ( A 
C_  RR  ->  A  e. 
_V )
1613, 15syl 16 . . . . . 6  |-  ( ph  ->  A  e.  _V )
17 eqidd 2444 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
18 eqidd 2444 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  =  ( x  e.  A  |->  ( B  -  C ) ) )
1916, 5, 7, 17, 18offval2 6357 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  ( B  -  ( B  -  C ) ) ) )
205, 6nncand 9745 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  ( B  -  C ) )  =  C )
2120mpteq2dva 4399 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  ( B  -  C )
) )  =  ( x  e.  A  |->  C ) )
2219, 21eqtrd 2475 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  C ) )
2322eleq1d 2509 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  oF  -  (
x  e.  A  |->  ( B  -  C ) ) )  e.  O(1)  <->  (
x  e.  A  |->  C )  e.  O(1) ) )
244, 23sylibd 214 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O(1)  ->  ( x  e.  A  |->  C )  e.  O(1) ) )
25 o1add 13112 . . . . 5  |-  ( ( ( x  e.  A  |->  ( B  -  C
) )  e.  O(1)  /\  ( x  e.  A  |->  C )  e.  O(1) )  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1) )
2625ex 434 . . . 4  |-  ( ( x  e.  A  |->  ( B  -  C ) )  e.  O(1)  ->  (
( x  e.  A  |->  C )  e.  O(1)  -> 
( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1) ) )
271, 26syl 16 . . 3  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e.  O(1)  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1) ) )
28 eqidd 2444 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
2916, 7, 6, 18, 28offval2 6357 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( ( B  -  C )  +  C ) ) )
305, 6npcand 9744 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  -  C
)  +  C )  =  B )
3130mpteq2dva 4399 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( ( B  -  C )  +  C
) )  =  ( x  e.  A  |->  B ) )
3229, 31eqtrd 2475 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  B ) )
3332eleq1d 2509 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1)  <->  (
x  e.  A  |->  B )  e.  O(1) ) )
3427, 33sylibd 214 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e.  O(1)  ->  ( x  e.  A  |->  B )  e.  O(1) ) )
3524, 34impbid 191 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 369    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993    C_ wss 3349    e. cmpt 4371   dom cdm 4861  (class class class)co 6112    oFcof 6339   CCcc 9301   RRcr 9302    + caddc 9306    - cmin 9616   O(1)co1 12985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ico 11327  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-o1 12989
This theorem is referenced by:  dchrmusum2  22765  dchrvmasumiflem2  22773  dchrisum0lem2a  22788  dchrisum0lem2  22789  rplogsum  22798  dirith2  22799  mulogsumlem  22802  mulogsum  22803  vmalogdivsum2  22809  vmalogdivsum  22810  2vmadivsumlem  22811  selberg3lem1  22828  selberg4lem1  22831  selberg4  22832  pntrlog2bndlem4  22851
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