Theorem elbigolo1 42351

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 x m 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 11843	. . . . . . . . . . . . 13 |- RR+ C_ RR
3	2	a1i 11	. . . . . . . . . . . 12 |- ( F : A --> RR+ -> RR+ C_ RR )
4	1, 3	fssd 6057	. . . . . . . . . . 11 |- ( F : A --> RR+ -> F : A --> RR )
5	4	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> F : A --> RR )
6	5	adantr 481	. . . . . . . . 9 |- ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) -> F : A --> RR )
7	6	ffvelrnda 6359	. . . . . . . 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 801	. . . . . . . 8 |- ( ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) /\ y e. A ) -> m e. RR )
9		simpl2 1065	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) -> G : A --> RR+ )
10	9	ffvelrnda 6359	. . . . . . . . 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 11878	. . . . . . . 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 1242	. . . . . . 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 10902	. . . . . . . 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 213	. . . . . . 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 6057	. . . . . . . . . . . 12 |- ( G : A --> RR+ -> G : A --> RR )
19	18	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> G : A --> RR )
20		reex 10027	. . . . . . . . . . . . 13 |- RR e. _V
21	20	ssex 4802	. . . . . . . . . . . 12 |- ( A C_ RR -> A e. _V )
22	21	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A e. _V )
23	5, 19, 22	3jca 1242	. . . . . . . . . 10 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( F : A --> RR /\ G : A --> RR /\ A e. _V ) )
24	23	adantr 481	. . . . . . . . 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 481	. . . . . . . 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 6048	. . . . . . . . . . . . . . . 16 |- ( G : A --> RR+ -> Fun G )
27	26	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( A C_ RR /\ G : A --> RR+ ) -> Fun G )
28	21	anim1i 592	. . . . . . . . . . . . . . . . 17 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( A e. _V /\ G : A --> RR+ ) )
29	28	ancomd 467	. . . . . . . . . . . . . . . 16 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G : A --> RR+ /\ A e. _V ) )
30		fex 6490	. . . . . . . . . . . . . . . 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 10041	. . . . . . . . . . . . . . 15 |- ( ( A C_ RR /\ G : A --> RR+ ) -> 0 e. RR )
33		frn 6053	. . . . . . . . . . . . . . . . 17 |- ( G : A --> RR+ -> ran G C_ RR+ )
34		0nrp 11865	. . . . . . . . . . . . . . . . . . 19 |- -. 0 e. RR+
35		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( ran G C_ RR+ -> ran G C_ RR+ )
36	35	ssneld 3605	. . . . . . . . . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . . . . 18 |- ( 0 e/ ran G <-> -. 0 e. ran G )
39	37, 38	sylibr 224	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 |- ( ( A C_ RR /\ G : A --> RR+ ) -> 0 e/ ran G )
42		suppdm 42300	. . . . . . . . . . . . . . 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 1329	. . . . . . . . . . . . . 14 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = dom G )
44		fdm 6051	. . . . . . . . . . . . . . 15 |- ( G : A --> RR+ -> dom G = A )
45	44	adantl 482	. . . . . . . . . . . . . 14 |- ( ( A C_ RR /\ G : A --> RR+ ) -> dom G = A )
46	43, 45	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = A )
47	46	3adant3 1081	. . . . . . . . . . . 12 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( G supp 0 ) = A )
48	47	eqcomd 2628	. . . . . . . . . . 11 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A = ( G supp 0 ) )
49	48	adantr 481	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) /\ ( x e. RR /\ m e. RR ) ) -> A = ( G supp 0 ) )
50	49	eleq2d 2687	. . . . . . . . 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 501	. . . . . . . 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 42340	. . . . . . . 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 693	. . . . . . 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 4663	. . . . . 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 271	. . . . 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 330	. . . 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 2985	. . 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 3056	. 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 1061	. . 3 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A C_ RR )
60		ssid 3624	. . . 4 |- A C_ A
61	60	a1i 11	. . 3 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A C_ A )
62		elbigo2 42346	. . 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 1327	. 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 42336	. . . . 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 2628	. . . . . . 7 |- ( G : A --> RR+ -> A = dom G )
67	66	3ad2ant2 1083	. . . . . 6 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A = dom G )
68		simpr 477	. . . . . . . . 9 |- ( ( A C_ RR /\ G : A --> RR+ ) -> G : A --> RR+ )
69	21	adantr 481	. . . . . . . . 9 |- ( ( A C_ RR /\ G : A --> RR+ ) -> A e. _V )
70	68, 69, 30	syl2anc 693	. . . . . . . 8 |- ( ( A C_ RR /\ G : A --> RR+ ) -> G e. _V )
71	27, 70, 32, 41, 42	syl31anc 1329	. . . . . . 7 |- ( ( A C_ RR /\ G : A --> RR+ ) -> ( G supp 0 ) = dom G )
72	71	3adant3 1081	. . . . . 6 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( G supp 0 ) = dom G )
73	67, 72	eqtr4d 2659	. . . . 5 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> A = ( G supp 0 ) )
74	73	feq2d 6031	. . . 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 247	. . 3 |- ( ( A C_ RR /\ G : A --> RR+ /\ F : A --> RR+ ) -> ( F /_f G ) : A --> RR )
76		ello12 14247	. . 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 693	. 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 300	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   RR+crp 11832   <_O(1)clo1 14218   /_f cfdiv 42331  _Ocbigo 42341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-supp 7296  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833  df-ico 12181  df-lo1 14222  df-fdiv 42332  df-bigo 42342

This theorem is referenced by: (None)