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Theorem o1dif 13091
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 13077 . . . . 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 9707 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
87ralrimiva 2789 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  ( B  -  C
)  e.  CC )
9 dmmptg 5323 . . . . . . . . 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 12992 . . . . . . . . 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 3379 . . . . . . 7  |-  ( ph  ->  A  C_  RR )
14 reex 9361 . . . . . . . 8  |-  RR  e.  _V
1514ssex 4424 . . . . . . 7  |-  ( A 
C_  RR  ->  A  e. 
_V )
1613, 15syl 16 . . . . . 6  |-  ( ph  ->  A  e.  _V )
17 eqidd 2434 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
18 eqidd 2434 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  =  ( x  e.  A  |->  ( B  -  C ) ) )
1916, 5, 7, 17, 18offval2 6325 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  ( B  -  ( B  -  C ) ) ) )
205, 6nncand 9712 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  ( B  -  C ) )  =  C )
2120mpteq2dva 4366 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  ( B  -  C )
) )  =  ( x  e.  A  |->  C ) )
2219, 21eqtrd 2465 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  C ) )
2322eleq1d 2499 . . 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 13075 . . . . 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 2434 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
2916, 7, 6, 18, 28offval2 6325 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( ( B  -  C )  +  C ) ) )
305, 6npcand 9711 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  -  C
)  +  C )  =  B )
3130mpteq2dva 4366 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( ( B  -  C )  +  C
) )  =  ( x  e.  A  |->  B ) )
3229, 31eqtrd 2465 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  B ) )
3332eleq1d 2499 . . 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 1362    e. wcel 1755   A.wral 2705   _Vcvv 2962    C_ wss 3316    e. cmpt 4338   dom cdm 4827  (class class class)co 6080    oFcof 6307   CCcc 9268   RRcr 9269    + caddc 9273    - cmin 9583   O(1)co1 12948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-ico 11294  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-o1 12952
This theorem is referenced by:  dchrmusum2  22628  dchrvmasumiflem2  22636  dchrisum0lem2a  22651  dchrisum0lem2  22652  rplogsum  22661  dirith2  22662  mulogsumlem  22665  mulogsum  22666  vmalogdivsum2  22672  vmalogdivsum  22673  2vmadivsumlem  22674  selberg3lem1  22691  selberg4lem1  22694  selberg4  22695  pntrlog2bndlem4  22714
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