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Theorem o1dif 13403
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 13389 . . . . 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 9921 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
87ralrimiva 2873 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  ( B  -  C
)  e.  CC )
9 dmmptg 5497 . . . . . . . . 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 13304 . . . . . . . . 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 3534 . . . . . . 7  |-  ( ph  ->  A  C_  RR )
14 reex 9574 . . . . . . . 8  |-  RR  e.  _V
1514ssex 4586 . . . . . . 7  |-  ( A 
C_  RR  ->  A  e. 
_V )
1613, 15syl 16 . . . . . 6  |-  ( ph  ->  A  e.  _V )
17 eqidd 2463 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
18 eqidd 2463 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  =  ( x  e.  A  |->  ( B  -  C ) ) )
1916, 5, 7, 17, 18offval2 6533 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  ( B  -  ( B  -  C ) ) ) )
205, 6nncand 9926 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  ( B  -  C ) )  =  C )
2120mpteq2dva 4528 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  ( B  -  C )
) )  =  ( x  e.  A  |->  C ) )
2219, 21eqtrd 2503 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  C ) )
2322eleq1d 2531 . . 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 13387 . . . . 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 2463 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
2916, 7, 6, 18, 28offval2 6533 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( ( B  -  C )  +  C ) ) )
305, 6npcand 9925 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  -  C
)  +  C )  =  B )
3130mpteq2dva 4528 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( ( B  -  C )  +  C
) )  =  ( x  e.  A  |->  B ) )
3229, 31eqtrd 2503 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  B ) )
3332eleq1d 2531 . . 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 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    C_ wss 3471    |-> cmpt 4500   dom cdm 4994  (class class class)co 6277    oFcof 6515   CCcc 9481   RRcr 9482    + caddc 9486    - cmin 9796   O(1)co1 13260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-ico 11526  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-o1 13264
This theorem is referenced by:  dchrmusum2  23402  dchrvmasumiflem2  23410  dchrisum0lem2a  23425  dchrisum0lem2  23426  rplogsum  23435  dirith2  23436  mulogsumlem  23439  mulogsum  23440  vmalogdivsum2  23446  vmalogdivsum  23447  2vmadivsumlem  23448  selberg3lem1  23465  selberg4lem1  23468  selberg4  23469  pntrlog2bndlem4  23488
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