Theorem elbigo 42345

Description: Properties of a function of order G(x). (Contributed by AV, 16-May-2020.)

Assertion

Ref	Expression
elbigo	|- ( F e. (_O ` G ) <-> ( F e. ( RR ^pm RR ) /\ G e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) )

Distinct variable groups:   x, G, m, y   m, F, x, y

Proof of Theorem elbigo

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bigoval 42343	. . . . 5 |- ( G e. ( RR ^pm RR ) -> (_O ` G ) = { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( G ` y ) ) } )
2	1	eleq2d 2687	. . . 4 |- ( G e. ( RR ^pm RR ) -> ( F e. (_O ` G ) <-> F e. { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( G ` y ) ) } ) )
3		dmeq 5324	. . . . . . . 8 |- ( f = F -> dom f = dom F )
4	3	ineq1d 3813	. . . . . . 7 |- ( f = F -> ( dom f i^i ( x [,) +oo ) ) = ( dom F i^i ( x [,) +oo ) ) )
5		fveq1 6190	. . . . . . . 8 |- ( f = F -> ( f ` y ) = ( F ` y ) )
6	5	breq1d 4663	. . . . . . 7 |- ( f = F -> ( ( f ` y ) <_ ( m x. ( G ` y ) ) <-> ( F ` y ) <_ ( m x. ( G ` y ) ) ) )
7	4, 6	raleqbidv 3152	. . . . . 6 |- ( f = F -> ( A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( G ` y ) ) <-> A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) )
8	7	2rexbidv 3057	. . . . 5 |- ( f = F -> ( E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( G ` y ) ) <-> E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) )
9	8	elrab 3363	. . . 4 |- ( F e. { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ ( m x. ( G ` y ) ) } <-> ( F e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) )
10	2, 9	syl6bb 276	. . 3 |- ( G e. ( RR ^pm RR ) -> ( F e. (_O ` G ) <-> ( F e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) ) )
11	10	pm5.32i 669	. 2 |- ( ( G e. ( RR ^pm RR ) /\ F e. (_O ` G ) ) <-> ( G e. ( RR ^pm RR ) /\ ( F e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) ) )
12		elbigofrcl 42344	. . 3 |- ( F e. (_O ` G ) -> G e. ( RR ^pm RR ) )
13	12	pm4.71ri 665	. 2 |- ( F e. (_O ` G ) <-> ( G e. ( RR ^pm RR ) /\ F e. (_O ` G ) ) )
14		3anan12 1051	. 2 |- ( ( F e. ( RR ^pm RR ) /\ G e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) <-> ( G e. ( RR ^pm RR ) /\ ( F e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) ) )
15	11, 13, 14	3bitr4i 292	1 |- ( F e. (_O ` G ) <-> ( F e. ( RR ^pm RR ) /\ G e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ ( m x. ( G ` y ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   RRcr 9935   x. cmul 9941   +oocpnf 10071   <_ cle 10075   [,)cico 12177  _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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-fv 5896  df-ov 6653  df-bigo 42342

This theorem is referenced by:  elbigo2  42346  elbigof  42348  elbigodm  42349