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Theorem elbigolo1 39712
Description: A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.)
Assertion
Ref Expression
elbigolo1  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F  e.  (_O `  G
)  <->  ( F /_f  G
)  e.  <_O(1) ) )

Proof of Theorem elbigolo1
Dummy variables  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 23 . . . . . . . . . . . 12  |-  ( F : A --> RR+  ->  F : A --> RR+ )
2 rpssre 11314 . . . . . . . . . . . . 13  |-  RR+  C_  RR
32a1i 11 . . . . . . . . . . . 12  |-  ( F : A --> RR+  ->  RR+  C_  RR )
41, 3fssd 5753 . . . . . . . . . . 11  |-  ( F : A --> RR+  ->  F : A --> RR )
543ad2ant3 1029 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  F : A --> RR )
65adantr 467 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  F : A
--> RR )
76ffvelrnda 6035 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( F `  y
)  e.  RR )
8 simplrr 770 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  m  e.  RR )
9 simpl2 1010 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  G : A
--> RR+ )
109ffvelrnda 6035 . . . . . . . . 9  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( G `  y
)  e.  RR+ )
1110rpregt0d 11349 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( G `  y )  e.  RR  /\  0  <  ( G `
 y ) ) )
127, 8, 113jca 1186 . . . . . . 7  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( F `  y )  e.  RR  /\  m  e.  RR  /\  ( ( G `  y )  e.  RR  /\  0  <  ( G `
 y ) ) ) )
13 ledivmul2 10486 . . . . . . . 8  |-  ( ( ( F `  y
)  e.  RR  /\  m  e.  RR  /\  (
( G `  y
)  e.  RR  /\  0  <  ( G `  y ) ) )  ->  ( ( ( F `  y )  /  ( G `  y ) )  <_  m 
<->  ( F `  y
)  <_  ( m  x.  ( G `  y
) ) ) )
1413bicomd 205 . . . . . . 7  |-  ( ( ( F `  y
)  e.  RR  /\  m  e.  RR  /\  (
( G `  y
)  e.  RR  /\  0  <  ( G `  y ) ) )  ->  ( ( F `
 y )  <_ 
( m  x.  ( G `  y )
)  <->  ( ( F `
 y )  / 
( G `  y
) )  <_  m
) )
1512, 14syl 17 . . . . . 6  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( F `  y )  <_  (
m  x.  ( G `
 y ) )  <-> 
( ( F `  y )  /  ( G `  y )
)  <_  m )
)
16 id 23 . . . . . . . . . . . . 13  |-  ( G : A --> RR+  ->  G : A --> RR+ )
172a1i 11 . . . . . . . . . . . . 13  |-  ( G : A --> RR+  ->  RR+  C_  RR )
1816, 17fssd 5753 . . . . . . . . . . . 12  |-  ( G : A --> RR+  ->  G : A --> RR )
19183ad2ant2 1028 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  G : A --> RR )
20 reex 9632 . . . . . . . . . . . . 13  |-  RR  e.  _V
2120ssex 4566 . . . . . . . . . . . 12  |-  ( A 
C_  RR  ->  A  e. 
_V )
22213ad2ant1 1027 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  e.  _V )
235, 19, 223jca 1186 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F : A --> RR  /\  G : A --> RR  /\  A  e.  _V )
)
2423adantr 467 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  ( F : A --> RR  /\  G : A --> RR  /\  A  e.  _V ) )
2524adantr 467 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( F : A --> RR  /\  G : A --> RR  /\  A  e.  _V ) )
26 ffun 5746 . . . . . . . . . . . . . . . 16  |-  ( G : A --> RR+  ->  Fun 
G )
2726adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  Fun  G )
2821anim1i 571 . . . . . . . . . . . . . . . . 17  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( A  e.  _V  /\  G : A --> RR+ ) )
2928ancomd 453 . . . . . . . . . . . . . . . 16  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G : A --> RR+  /\  A  e.  _V ) )
30 fex 6151 . . . . . . . . . . . . . . . 16  |-  ( ( G : A --> RR+  /\  A  e.  _V )  ->  G  e.  _V )
3129, 30syl 17 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  G  e.  _V )
32 0red 9646 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  0  e.  RR )
33 frn 5750 . . . . . . . . . . . . . . . . 17  |-  ( G : A --> RR+  ->  ran 
G  C_  RR+ )
34 0nrp 11336 . . . . . . . . . . . . . . . . . . 19  |-  -.  0  e.  RR+
35 id 23 . . . . . . . . . . . . . . . . . . . 20  |-  ( ran 
G  C_  RR+  ->  ran  G 
C_  RR+ )
3635ssneld 3467 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
G  C_  RR+  ->  ( -.  0  e.  RR+  ->  -.  0  e.  ran  G
) )
3734, 36mpi 21 . . . . . . . . . . . . . . . . . 18  |-  ( ran 
G  C_  RR+  ->  -.  0  e.  ran  G )
38 df-nel 2622 . . . . . . . . . . . . . . . . . 18  |-  ( 0  e/  ran  G  <->  -.  0  e.  ran  G )
3937, 38sylibr 216 . . . . . . . . . . . . . . . . 17  |-  ( ran 
G  C_  RR+  ->  0  e/  ran  G )
4033, 39syl 17 . . . . . . . . . . . . . . . 16  |-  ( G : A --> RR+  ->  0  e/  ran  G )
4140adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  0  e/  ran  G )
42 suppdm 39648 . . . . . . . . . . . . . . 15  |-  ( ( ( Fun  G  /\  G  e.  _V  /\  0  e.  RR )  /\  0  e/  ran  G )  -> 
( G supp  0 )  =  dom  G )
4327, 31, 32, 41, 42syl31anc 1268 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  dom  G )
44 fdm 5748 . . . . . . . . . . . . . . 15  |-  ( G : A --> RR+  ->  dom 
G  =  A )
4544adantl 468 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  dom  G  =  A )
4643, 45eqtrd 2464 . . . . . . . . . . . . 13  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  A )
47463adant3 1026 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( G supp  0 )  =  A )
4847eqcomd 2431 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  =  ( G supp  0
) )
4948adantr 467 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  A  =  ( G supp  0 )
)
5049eleq2d 2493 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  ( y  e.  A  <->  y  e.  ( G supp  0 ) ) )
5150biimpa 487 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  y  e.  ( G supp  0 ) )
52 refdivmptfv 39701 . . . . . . . 8  |-  ( ( ( F : A --> RR  /\  G : A --> RR  /\  A  e.  _V )  /\  y  e.  ( G supp  0 ) )  ->  ( ( F /_f 
G ) `  y
)  =  ( ( F `  y )  /  ( G `  y ) ) )
5325, 51, 52syl2anc 666 . . . . . . 7  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( F /_f  G
) `  y )  =  ( ( F `
 y )  / 
( G `  y
) ) )
5453breq1d 4431 . . . . . 6  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( ( F /_f 
G ) `  y
)  <_  m  <->  ( ( F `  y )  /  ( G `  y ) )  <_  m ) )
5515, 54bitr4d 260 . . . . 5  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( F `  y )  <_  (
m  x.  ( G `
 y ) )  <-> 
( ( F /_f  G
) `  y )  <_  m ) )
5655imbi2d 318 . . . 4  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( ( x  <_ 
y  ->  ( F `  y )  <_  (
m  x.  ( G `
 y ) ) )  <->  ( x  <_ 
y  ->  ( ( F /_f  G ) `  y
)  <_  m )
) )
5756ralbidva 2862 . . 3  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  ( A. y  e.  A  (
x  <_  y  ->  ( F `  y )  <_  ( m  x.  ( G `  y
) ) )  <->  A. y  e.  A  ( x  <_  y  ->  ( ( F /_f  G ) `  y
)  <_  m )
) )
58572rexbidva 2946 . 2  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( E. x  e.  RR  E. m  e.  RR  A. y  e.  A  (
x  <_  y  ->  ( F `  y )  <_  ( m  x.  ( G `  y
) ) )  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  ( x  <_ 
y  ->  ( ( F /_f  G ) `  y
)  <_  m )
) )
59 simp1 1006 . . 3  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  C_  RR )
60 ssid 3484 . . . 4  |-  A  C_  A
6160a1i 11 . . 3  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  C_  A )
62 elbigo2 39707 . . 3  |-  ( ( ( G : A --> RR  /\  A  C_  RR )  /\  ( F : A
--> RR  /\  A  C_  A ) )  -> 
( F  e.  (_O
`  G )  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  ( x  <_ 
y  ->  ( F `  y )  <_  (
m  x.  ( G `
 y ) ) ) ) )
6319, 59, 5, 61, 62syl22anc 1266 . 2  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F  e.  (_O `  G
)  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  (
x  <_  y  ->  ( F `  y )  <_  ( m  x.  ( G `  y
) ) ) ) )
64 refdivmptf 39697 . . . . 5  |-  ( ( F : A --> RR  /\  G : A --> RR  /\  A  e.  _V )  ->  ( F /_f  G ) : ( G supp  0
) --> RR )
6523, 64syl 17 . . . 4  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F /_f  G ) : ( G supp  0 ) --> RR )
6644eqcomd 2431 . . . . . . 7  |-  ( G : A --> RR+  ->  A  =  dom  G )
67663ad2ant2 1028 . . . . . 6  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  =  dom  G )
68 simpr 463 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  G : A --> RR+ )
6921adantr 467 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  A  e.  _V )
7068, 69, 30syl2anc 666 . . . . . . . 8  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  G  e.  _V )
7127, 70, 32, 41, 42syl31anc 1268 . . . . . . 7  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  dom  G )
72713adant3 1026 . . . . . 6  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( G supp  0 )  =  dom  G )
7367, 72eqtr4d 2467 . . . . 5  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  =  ( G supp  0
) )
7473feq2d 5731 . . . 4  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  (
( F /_f  G ) : A --> RR  <->  ( F /_f  G ) : ( G supp  0 ) --> RR ) )
7565, 74mpbird 236 . . 3  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F /_f  G ) : A --> RR )
76 ello12 13573 . . 3  |-  ( ( ( F /_f  G ) : A --> RR  /\  A  C_  RR )  -> 
( ( F /_f  G
)  e.  <_O(1)  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  ( x  <_ 
y  ->  ( ( F /_f  G ) `  y
)  <_  m )
) )
7775, 59, 76syl2anc 666 . 2  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  (
( F /_f  G )  e.  <_O(1)  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  (
x  <_  y  ->  ( ( F /_f  G ) `
 y )  <_  m ) ) )
7858, 63, 773bitr4d 289 1  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( F  e.  (_O `  G
)  <->  ( F /_f  G
)  e.  <_O(1) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    e/ wnel 2620   A.wral 2776   E.wrex 2777   _Vcvv 3082    C_ wss 3437   class class class wbr 4421   dom cdm 4851   ran crn 4852   Fun wfun 5593   -->wf 5595   ` cfv 5599  (class class class)co 6303   supp csupp 6923   RRcr 9540   0cc0 9541    x. cmul 9546    < clt 9677    <_ cle 9678    / cdiv 10271   RR+crp 11304   <_O(1)clo1 13544   /_f cfdiv 39692  _Ocbigo 39702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-supp 6924  df-er 7369  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-rp 11305  df-ico 11643  df-lo1 13548  df-fdiv 39693  df-bigo 39703
This theorem is referenced by: (None)
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