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Theorem o1dif 13692
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 13678 . . . . 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 436 . . . 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 17 . . 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 9993 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
87ralrimiva 2836 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  ( B  -  C
)  e.  CC )
9 dmmptg 5351 . . . . . . . . 9  |-  ( A. x  e.  A  ( B  -  C )  e.  CC  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  =  A )
108, 9syl 17 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  =  A )
11 o1dm 13593 . . . . . . . . 9  |-  ( ( x  e.  A  |->  ( B  -  C ) )  e.  O(1)  ->  dom  ( x  e.  A  |->  ( B  -  C
) )  C_  RR )
121, 11syl 17 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  A  |->  ( B  -  C ) )  C_  RR )
1310, 12eqsstr3d 3499 . . . . . . 7  |-  ( ph  ->  A  C_  RR )
14 reex 9637 . . . . . . . 8  |-  RR  e.  _V
1514ssex 4568 . . . . . . 7  |-  ( A 
C_  RR  ->  A  e. 
_V )
1613, 15syl 17 . . . . . 6  |-  ( ph  ->  A  e.  _V )
17 eqidd 2423 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
18 eqidd 2423 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  =  ( x  e.  A  |->  ( B  -  C ) ) )
1916, 5, 7, 17, 18offval2 6562 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  ( B  -  ( B  -  C ) ) ) )
205, 6nncand 9998 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  ( B  -  C ) )  =  C )
2120mpteq2dva 4510 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  ( B  -  C )
) )  =  ( x  e.  A  |->  C ) )
2219, 21eqtrd 2463 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  C ) )
2322eleq1d 2491 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  oF  -  (
x  e.  A  |->  ( B  -  C ) ) )  e.  O(1)  <->  (
x  e.  A  |->  C )  e.  O(1) ) )
244, 23sylibd 217 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O(1)  ->  ( x  e.  A  |->  C )  e.  O(1) ) )
25 o1add 13676 . . . . 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 435 . . . 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 17 . . 3  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e.  O(1)  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1) ) )
28 eqidd 2423 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
2916, 7, 6, 18, 28offval2 6562 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( ( B  -  C )  +  C ) ) )
305, 6npcand 9997 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  -  C
)  +  C )  =  B )
3130mpteq2dva 4510 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( ( B  -  C )  +  C
) )  =  ( x  e.  A  |->  B ) )
3229, 31eqtrd 2463 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  B ) )
3332eleq1d 2491 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  e.  O(1)  <->  (
x  e.  A  |->  B )  e.  O(1) ) )
3427, 33sylibd 217 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  e.  O(1)  ->  ( x  e.  A  |->  B )  e.  O(1) ) )
3524, 34impbid 193 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436    |-> cmpt 4482   dom cdm 4853  (class class class)co 6305    oFcof 6543   CCcc 9544   RRcr 9545    + caddc 9549    - cmin 9867   O(1)co1 13549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-o1 13553
This theorem is referenced by:  dchrmusum2  24330  dchrvmasumiflem2  24338  dchrisum0lem2a  24353  dchrisum0lem2  24354  rplogsum  24363  dirith2  24364  mulogsumlem  24367  mulogsum  24368  vmalogdivsum2  24374  vmalogdivsum  24375  2vmadivsumlem  24376  selberg3lem1  24393  selberg4lem1  24396  selberg4  24397  pntrlog2bndlem4  24416
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