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.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . . . 12 |- ( F : A --> RR+ -> F : A --> RR+ )
2		rpssre 11312	. . . . . . . . . . . . 13 |- RR+ C_ RR
3	2	a1i 11	. . . . . . . . . . . 12 |- ( F : A --> RR+ -> RR+ C_ RR )
4	1, 3	fssd 5738	. . . . . . . . . . 11 |- ( F : A --> RR+ -> F : A --> RR )
5	4	3ad2ant3 1031	. . . . . . . . . 10 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> F : A --> RR )
6	5	adantr 467	. . . . . . . . 9 |- ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) -> F : A --> RR )
7	6	ffvelrnda 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+ )
10	9	ffvelrnda 6022	. . . . . . . . 9 |- ( ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) /\ y e. A ) -> ( G ` y ) e. RR+ )
11	10	rpregt0d 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 ) ) )
12	7, 8, 11	3jca 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 ) ) ) )
14	13	bicomd 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 ) )
15	12, 14	syl 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+ )
17	2	a1i 11	. . . . . . . . . . . . 13 |- ( G : A --> RR+ -> RR+ C_ RR )
18	16, 17	fssd 5738	. . . . . . . . . . . 12 |- ( G : A --> RR+ -> G : A --> RR )
19	18	3ad2ant2 1030	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> G : A --> RR )
20		reex 9630	. . . . . . . . . . . . 13 |- RR e. _V
21	20	ssex 4547	. . . . . . . . . . . 12 |- ( A C_ RR -> A e. _V )
22	21	3ad2ant1 1029	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A e. _V )
23	5, 19, 22	3jca 1188	. . . . . . . . . 10 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( F : A --> RR /\ G : A --> RR /\ A e. _V ) )
24	23	adantr 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 ) )
25	24	adantr 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 )
27	26	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( A C_ RR /\ G : A --> RR+ ) -> Fun G )
28	21	anim1i 572	. . . . . . . . . . . . . . . . 17 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( A e. _V /\ G : A --> RR+ ) )
29	28	ancomd 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 )
31	29, 30	syl 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+ )
36	35	ssneld 3434	. . . . . . . . . . . . . . . . . . 19 |- ( ran G C_ RR+ -> ( -. 0 e. RR+ -> -. 0 e. ran G ) )
37	34, 36	mpi 20	. . . . . . . . . . . . . . . . . 18 |- ( ran G C_ RR+ -> -. 0 e. ran G )
38		df-nel 2625	. . . . . . . . . . . . . . . . . 18 |- ( 0 e/ ran G <-> -. 0 e. ran G )
39	37, 38	sylibr 216	. . . . . . . . . . . . . . . . 17 |- ( ran G C_ RR+ -> 0 e/ ran G )
40	33, 39	syl 17	. . . . . . . . . . . . . . . 16 |- ( G : A --> RR+ -> 0 e/ ran G )
41	40	adantl 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 )
43	27, 31, 32, 41, 42	syl31anc 1271	. . . . . . . . . . . . . 14 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = dom G )
44		fdm 5733	. . . . . . . . . . . . . . 15 |- ( G : A --> RR+ -> dom G = A )
45	44	adantl 468	. . . . . . . . . . . . . 14 |- ( ( A C_ RR /\ G : A --> RR+ ) -> dom G = A )
46	43, 45	eqtrd 2485	. . . . . . . . . . . . 13 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = A )
47	46	3adant3 1028	. . . . . . . . . . . 12 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( G supp 0 ) = A )
48	47	eqcomd 2457	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A = ( G supp 0 ) )
49	48	adantr 467	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) -> A = ( G supp 0 ) )
50	49	eleq2d 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 ) ) )
51	50	biimpa 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 ) ) )
53	25, 51, 52	syl2anc 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 ) ) )
54	53	breq1d 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 ) )
55	15, 54	bitr4d 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 ) )
56	55	imbi2d 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 ) ) )
57	56	ralbidva 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 ) ) )
58	57	2rexbidva 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
61	60	a1i 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 ) ) ) ) )
63	19, 59, 5, 61, 62	syl22anc 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 )
65	23, 64	syl 17	. . . 4 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( F /_f G ) : ( G supp 0 ) --> RR )
66	44	eqcomd 2457	. . . . . . 7 |- ( G : A --> RR+ -> A = dom G )
67	66	3ad2ant2 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+ )
69	21	adantr 467	. . . . . . . . 9 |- ( ( A C_ RR /\ G : A --> RR+ ) -> A e. _V )
70	68, 69, 30	syl2anc 667	. . . . . . . 8 |- ( ( A C_ RR /\ G : A --> RR+ ) -> G e. _V )
71	27, 70, 32, 41, 42	syl31anc 1271	. . . . . . 7 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = dom G )
72	71	3adant3 1028	. . . . . 6 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( G supp 0 ) = dom G )
73	67, 72	eqtr4d 2488	. . . . 5 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A = ( G supp 0 ) )
74	73	feq2d 5715	. . . 4 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( ( F /_f G ) : A --> RR <-> ( F /_f G ) : ( G supp 0 ) --> RR ) )
75	65, 74	mpbird 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 ) ) )
77	75, 59, 76	syl2anc 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 ) ) )
78	58, 63, 77	3bitr4d 289	1 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( F e. (_O ` G ) <-> ( F /_f G ) e. <_O(1) ) )

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)