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Theorem elbigolo1 40421
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 22 . . . . . . . . . . . 12  |-  ( F : A --> RR+  ->  F : A --> RR+ )
2 rpssre 11312 . . . . . . . . . . . . 13  |-  RR+  C_  RR
32a1i 11 . . . . . . . . . . . 12  |-  ( F : A --> RR+  ->  RR+  C_  RR )
41, 3fssd 5738 . . . . . . . . . . 11  |-  ( F : A --> RR+  ->  F : A --> RR )
543ad2ant3 1031 . . . . . . . . . 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 6022 . . . . . . . 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 771 . . . . . . . 8  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  m  e.  RR )
9 simpl2 1012 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A --> RR+ )  /\  (
x  e.  RR  /\  m  e.  RR )
)  ->  G : A
--> RR+ )
109ffvelrnda 6022 . . . . . . . . 9  |-  ( ( ( ( A  C_  RR  /\  G : A --> RR+ 
/\  F : A --> RR+ )  /\  ( x  e.  RR  /\  m  e.  RR ) )  /\  y  e.  A )  ->  ( G `  y
)  e.  RR+ )
1110rpregt0d 11347 . . . . . . . 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 1188 . . . . . . 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 10484 . . . . . . . 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 22 . . . . . . . . . . . . 13  |-  ( G : A --> RR+  ->  G : A --> RR+ )
172a1i 11 . . . . . . . . . . . . 13  |-  ( G : A --> RR+  ->  RR+  C_  RR )
1816, 17fssd 5738 . . . . . . . . . . . 12  |-  ( G : A --> RR+  ->  G : A --> RR )
19183ad2ant2 1030 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  G : A --> RR )
20 reex 9630 . . . . . . . . . . . . 13  |-  RR  e.  _V
2120ssex 4547 . . . . . . . . . . . 12  |-  ( A 
C_  RR  ->  A  e. 
_V )
22213ad2ant1 1029 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  e.  _V )
235, 19, 223jca 1188 . . . . . . . . . 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 5731 . . . . . . . . . . . . . . . 16  |-  ( G : A --> RR+  ->  Fun 
G )
2726adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  Fun  G )
2821anim1i 572 . . . . . . . . . . . . . . . . 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 6138 . . . . . . . . . . . . . . . 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 9644 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  0  e.  RR )
33 frn 5735 . . . . . . . . . . . . . . . . 17  |-  ( G : A --> RR+  ->  ran 
G  C_  RR+ )
34 0nrp 11334 . . . . . . . . . . . . . . . . . . 19  |-  -.  0  e.  RR+
35 id 22 . . . . . . . . . . . . . . . . . . . 20  |-  ( ran 
G  C_  RR+  ->  ran  G 
C_  RR+ )
3635ssneld 3434 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
G  C_  RR+  ->  ( -.  0  e.  RR+  ->  -.  0  e.  ran  G
) )
3734, 36mpi 20 . . . . . . . . . . . . . . . . . 18  |-  ( ran 
G  C_  RR+  ->  -.  0  e.  ran  G )
38 df-nel 2625 . . . . . . . . . . . . . . . . . 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 40357 . . . . . . . . . . . . . . 15  |-  ( ( ( Fun  G  /\  G  e.  _V  /\  0  e.  RR )  /\  0  e/  ran  G )  -> 
( G supp  0 )  =  dom  G )
4327, 31, 32, 41, 42syl31anc 1271 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  dom  G )
44 fdm 5733 . . . . . . . . . . . . . . 15  |-  ( G : A --> RR+  ->  dom 
G  =  A )
4544adantl 468 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  dom  G  =  A )
4643, 45eqtrd 2485 . . . . . . . . . . . . 13  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  A )
47463adant3 1028 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( G supp  0 )  =  A )
4847eqcomd 2457 . . . . . . . . . . 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 2514 . . . . . . . . 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 40410 . . . . . . . 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 667 . . . . . . 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 4412 . . . . . 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 2824 . . 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 2907 . 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 1008 . . 3  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  C_  RR )
60 ssid 3451 . . . 4  |-  A  C_  A
6160a1i 11 . . 3  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  C_  A )
62 elbigo2 40416 . . 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 1269 . 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 40406 . . . . 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 2457 . . . . . . 7  |-  ( G : A --> RR+  ->  A  =  dom  G )
67663ad2ant2 1030 . . . . . 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 667 . . . . . . . 8  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  G  e.  _V )
7127, 70, 32, 41, 42syl31anc 1271 . . . . . . 7  |-  ( ( A  C_  RR  /\  G : A --> RR+ )  ->  ( G supp  0 )  =  dom  G )
72713adant3 1028 . . . . . 6  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  ( G supp  0 )  =  dom  G )
7367, 72eqtr4d 2488 . . . . 5  |-  ( ( A  C_  RR  /\  G : A --> RR+  /\  F : A
--> RR+ )  ->  A  =  ( G supp  0
) )
7473feq2d 5715 . . . 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 13580 . . 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 667 . 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 985    = wceq 1444    e. wcel 1887    e/ wnel 2623   A.wral 2737   E.wrex 2738   _Vcvv 3045    C_ wss 3404   class class class wbr 4402   dom cdm 4834   ran crn 4835   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   supp csupp 6914   RRcr 9538   0cc0 9539    x. cmul 9544    < clt 9675    <_ cle 9676    / cdiv 10269   RR+crp 11302   <_O(1)clo1 13551   /_f cfdiv 40401  _Ocbigo 40411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-supp 6915  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-rp 11303  df-ico 11641  df-lo1 13555  df-fdiv 40402  df-bigo 40412
This theorem is referenced by: (None)
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