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Theorem o1dif 13599
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 13585 . . . . 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 433 . . . 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 9966 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
87ralrimiva 2817 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  ( B  -  C
)  e.  CC )
9 dmmptg 5319 . . . . . . . . 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 13500 . . . . . . . . 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 3476 . . . . . . 7  |-  ( ph  ->  A  C_  RR )
14 reex 9612 . . . . . . . 8  |-  RR  e.  _V
1514ssex 4537 . . . . . . 7  |-  ( A 
C_  RR  ->  A  e. 
_V )
1613, 15syl 17 . . . . . 6  |-  ( ph  ->  A  e.  _V )
17 eqidd 2403 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
18 eqidd 2403 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  =  ( x  e.  A  |->  ( B  -  C ) ) )
1916, 5, 7, 17, 18offval2 6537 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  ( B  -  ( B  -  C ) ) ) )
205, 6nncand 9971 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  ( B  -  C ) )  =  C )
2120mpteq2dva 4480 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  ( B  -  C )
) )  =  ( x  e.  A  |->  C ) )
2219, 21eqtrd 2443 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B )  oF  -  ( x  e.  A  |->  ( B  -  C ) ) )  =  ( x  e.  A  |->  C ) )
2322eleq1d 2471 . . 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 13583 . . . . 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 432 . . . 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 2403 . . . . . 6  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
2916, 7, 6, 18, 28offval2 6537 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( ( B  -  C )  +  C ) ) )
305, 6npcand 9970 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( B  -  C
)  +  C )  =  B )
3130mpteq2dva 4480 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  ( ( B  -  C )  +  C
) )  =  ( x  e.  A  |->  B ) )
3229, 31eqtrd 2443 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  ( B  -  C ) )  oF  +  ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  B ) )
3332eleq1d 2471 . . 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058    C_ wss 3413    |-> cmpt 4452   dom cdm 4822  (class class class)co 6277    oFcof 6518   CCcc 9519   RRcr 9520    + caddc 9524    - cmin 9840   O(1)co1 13456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ico 11587  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-o1 13460
This theorem is referenced by:  dchrmusum2  24058  dchrvmasumiflem2  24066  dchrisum0lem2a  24081  dchrisum0lem2  24082  rplogsum  24091  dirith2  24092  mulogsumlem  24095  mulogsum  24096  vmalogdivsum2  24102  vmalogdivsum  24103  2vmadivsumlem  24104  selberg3lem1  24121  selberg4lem1  24124  selberg4  24125  pntrlog2bndlem4  24144
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