Theorem amgm 24717

Description: Inequality of arithmetic and geometric means. Here ( M gsum_g F ) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements F ( x ) , x e. A together), and (ℂ_fld gsum_g F ) calculates the group sum in the additive group (i.e. the sum of the elements). This is Metamath 100 proof #38. (Contributed by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
amgm.1	|- M = (mulGrp ` ℂfld )

Assertion

Ref	Expression
amgm	|- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

Proof of Theorem amgm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		amgm.1	. . . . . . . . 9 |- M = (mulGrp ` ℂfld )
2		cnfldbas 19750	. . . . . . . . 9 |- CC = ( Base ` ℂfld )
3	1, 2	mgpbas 18495	. . . . . . . 8 |- CC = ( Base ` M )
4		cnfld1 19771	. . . . . . . . 9 |- 1 = ( 1r ` ℂfld )
5	1, 4	ringidval 18503	. . . . . . . 8 |- 1 = ( 0g ` M )
6		cnfldmul 19752	. . . . . . . . 9 |- x. = ( .r ` ℂfld )
7	1, 6	mgpplusg 18493	. . . . . . . 8 |- x. = ( +g ` M )
8		cncrng 19767	. . . . . . . . 9 |- ℂfld e. CRing
9	1	crngmgp 18555	. . . . . . . . 9 |- (ℂfld e. CRing -> M e. CMnd )
10	8, 9	mp1i 13	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> M e. CMnd )
11		simpl1 1064	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> A e. Fin )
12		simpl3 1066	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> F : A --> ( 0 [,) +oo ) )
13		rge0ssre 12280	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
14		ax-resscn 9993	. . . . . . . . . 10 |- RR C_ CC
15	13, 14	sstri 3612	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ CC
16		fss 6056	. . . . . . . . 9 |- ( ( F : A --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> F : A --> CC )
17	12, 15, 16	sylancl 694	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> F : A --> CC )
18		1ex 10035	. . . . . . . . . 10 |- 1 e. _V
19	18	a1i 11	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> 1 e. _V )
20	17, 11, 19	fdmfifsupp 8285	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> F finSupp 1 )
21		disjdif 4040	. . . . . . . . 9 |- ( { x } i^i ( A \ { x } ) ) = (/)
22	21	a1i 11	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( { x } i^i ( A \ { x } ) ) = (/) )
23		undif2 4044	. . . . . . . . 9 |- ( { x } u. ( A \ { x } ) ) = ( { x } u. A )
24		simprl 794	. . . . . . . . . . 11 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> x e. A )
25	24	snssd 4340	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> { x } C_ A )
26		ssequn1 3783	. . . . . . . . . 10 |- ( { x } C_ A <-> ( { x } u. A ) = A )
27	25, 26	sylib 208	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( { x } u. A ) = A )
28	23, 27	syl5req 2669	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> A = ( { x } u. ( A \ { x } ) ) )
29	3, 5, 7, 10, 11, 17, 20, 22, 28	gsumsplit 18328	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg F ) = ( ( M gsumg ( F |` { x } ) ) x. ( M gsumg ( F |` ( A \ { x } ) ) ) ) )
30	12, 25	feqresmpt 6250	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( F |` { x } ) = ( y e. { x } |-> ( F ` y ) ) )
31	30	oveq2d 6666	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg ( F |` { x } ) ) = ( M gsumg ( y e. { x } |-> ( F ` y ) ) ) )
32		cnring 19768	. . . . . . . . . . 11 |- ℂfld e. Ring
33	1	ringmgp 18553	. . . . . . . . . . 11 |- (ℂfld e. Ring -> M e. Mnd )
34	32, 33	mp1i 13	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> M e. Mnd )
35	17, 24	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( F ` x ) e. CC )
36		fveq2 6191	. . . . . . . . . . 11 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	3, 36	gsumsn 18354	. . . . . . . . . 10 |- ( ( M e. Mnd /\ x e. A /\ ( F ` x ) e. CC ) -> ( M gsumg ( y e. { x } |-> ( F ` y ) ) ) = ( F ` x ) )
38	34, 24, 35, 37	syl3anc 1326	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg ( y e. { x } |-> ( F ` y ) ) ) = ( F ` x ) )
39		simprr 796	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( F ` x ) = 0 )
40	31, 38, 39	3eqtrd 2660	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg ( F |` { x } ) ) = 0 )
41	40	oveq1d 6665	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( ( M gsumg ( F |` { x } ) ) x. ( M gsumg ( F |` ( A \ { x } ) ) ) ) = ( 0 x. ( M gsumg ( F |` ( A \ { x } ) ) ) ) )
42		diffi 8192	. . . . . . . . . 10 |- ( A e. Fin -> ( A \ { x } ) e. Fin )
43	11, 42	syl 17	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( A \ { x } ) e. Fin )
44		difss 3737	. . . . . . . . . 10 |- ( A \ { x } ) C_ A
45		fssres 6070	. . . . . . . . . 10 |- ( ( F : A --> CC /\ ( A \ { x } ) C_ A ) -> ( F |` ( A \ { x } ) ) : ( A \ { x } ) --> CC )
46	17, 44, 45	sylancl 694	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( F |` ( A \ { x } ) ) : ( A \ { x } ) --> CC )
47	46, 43, 19	fdmfifsupp 8285	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( F |` ( A \ { x } ) ) finSupp 1 )
48	3, 5, 10, 43, 46, 47	gsumcl 18316	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg ( F |` ( A \ { x } ) ) ) e. CC )
49	48	mul02d 10234	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( 0 x. ( M gsumg ( F |` ( A \ { x } ) ) ) ) = 0 )
50	29, 41, 49	3eqtrd 2660	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( M gsumg F ) = 0 )
51	50	oveq1d 6665	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) = ( 0 ^c ( 1 / ( # ` A ) ) ) )
52		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> A =/= (/) )
53		hashnncl 13157	. . . . . . . . . 10 |- ( A e. Fin -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
54	11, 53	syl 17	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
55	52, 54	mpbird 247	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( # ` A ) e. NN )
56	55	nncnd 11036	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( # ` A ) e. CC )
57	55	nnne0d 11065	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( # ` A ) =/= 0 )
58	56, 57	reccld 10794	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( 1 / ( # ` A ) ) e. CC )
59	56, 57	recne0d 10795	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( 1 / ( # ` A ) ) =/= 0 )
60	58, 59	0cxpd 24456	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( 0 ^c ( 1 / ( # ` A ) ) ) = 0 )
61	51, 60	eqtrd 2656	. . . 4 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) = 0 )
62		cnfld0 19770	. . . . . . 7 |- 0 = ( 0g ` ℂfld )
63		ringcmn 18581	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. CMnd )
64	32, 63	mp1i 13	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ℂfld e. CMnd )
65		rege0subm 19802	. . . . . . . 8 |- ( 0 [,) +oo ) e. (SubMnd ` ℂfld )
66	65	a1i 11	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( 0 [,) +oo ) e. (SubMnd ` ℂfld ) )
67		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
68	67	a1i 11	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> 0 e. _V )
69	12, 11, 68	fdmfifsupp 8285	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> F finSupp 0 )
70	62, 64, 11, 66, 12, 69	gsumsubmcl 18319	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> (ℂfld gsumg F ) e. ( 0 [,) +oo ) )
71		elrege0 12278	. . . . . 6 |- ( (ℂfld gsumg F ) e. ( 0 [,) +oo ) <-> ( (ℂfld gsumg F ) e. RR /\ 0 <_ (ℂfld gsumg F ) ) )
72	70, 71	sylib 208	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( (ℂfld gsumg F ) e. RR /\ 0 <_ (ℂfld gsumg F ) ) )
73	55	nnred 11035	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( # ` A ) e. RR )
74	55	nngt0d 11064	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> 0 < ( # ` A ) )
75		divge0 10892	. . . . 5 |- ( ( ( (ℂfld gsumg F ) e. RR /\ 0 <_ (ℂfld gsumg F ) ) /\ ( ( # ` A ) e. RR /\ 0 < ( # ` A ) ) ) -> 0 <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )
76	72, 73, 74, 75	syl12anc 1324	. . . 4 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> 0 <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )
77	61, 76	eqbrtrd 4675	. . 3 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. A /\ ( F ` x ) = 0 ) ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )
78	77	rexlimdvaa 3032	. 2 |- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( E. x e. A ( F ` x ) = 0 -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
79		ralnex 2992	. . 3 |- ( A. x e. A -. ( F ` x ) = 0 <-> -. E. x e. A ( F ` x ) = 0 )
80		simpl1 1064	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> A e. Fin )
81		simpl2 1065	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> A =/= (/) )
82		simpl3 1066	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> F : A --> ( 0 [,) +oo ) )
83		ffn 6045	. . . . . . 7 |- ( F : A --> ( 0 [,) +oo ) -> F Fn A )
84	82, 83	syl 17	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> F Fn A )
85		ffvelrn 6357	. . . . . . . . . . . . . . . 16 |- ( ( F : A --> ( 0 [,) +oo ) /\ x e. A ) -> ( F ` x ) e. ( 0 [,) +oo ) )
86	85	3ad2antl3 1225	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( F ` x ) e. ( 0 [,) +oo ) )
87		elrege0 12278	. . . . . . . . . . . . . . 15 |- ( ( F ` x ) e. ( 0 [,) +oo ) <-> ( ( F ` x ) e. RR /\ 0 <_ ( F ` x ) ) )
88	86, 87	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( ( F ` x ) e. RR /\ 0 <_ ( F ` x ) ) )
89	88	simprd 479	. . . . . . . . . . . . 13 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> 0 <_ ( F ` x ) )
90		0re 10040	. . . . . . . . . . . . . 14 |- 0 e. RR
91	88	simpld 475	. . . . . . . . . . . . . 14 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( F ` x ) e. RR )
92		leloe 10124	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR /\ ( F ` x ) e. RR ) -> ( 0 <_ ( F ` x ) <-> ( 0 < ( F ` x ) \/ 0 = ( F ` x ) ) ) )
93	90, 91, 92	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( 0 <_ ( F ` x ) <-> ( 0 < ( F ` x ) \/ 0 = ( F ` x ) ) ) )
94	89, 93	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( 0 < ( F ` x ) \/ 0 = ( F ` x ) ) )
95	94	ord 392	. . . . . . . . . . 11 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( -. 0 < ( F ` x ) -> 0 = ( F ` x ) ) )
96		eqcom 2629	. . . . . . . . . . 11 |- ( 0 = ( F ` x ) <-> ( F ` x ) = 0 )
97	95, 96	syl6ib 241	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( -. 0 < ( F ` x ) -> ( F ` x ) = 0 ) )
98	97	con1d 139	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( -. ( F ` x ) = 0 -> 0 < ( F ` x ) ) )
99		elrp 11834	. . . . . . . . . . 11 |- ( ( F ` x ) e. RR+ <-> ( ( F ` x ) e. RR /\ 0 < ( F ` x ) ) )
100	99	baib 944	. . . . . . . . . 10 |- ( ( F ` x ) e. RR -> ( ( F ` x ) e. RR+ <-> 0 < ( F ` x ) ) )
101	91, 100	syl 17	. . . . . . . . 9 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( ( F ` x ) e. RR+ <-> 0 < ( F ` x ) ) )
102	98, 101	sylibrd 249	. . . . . . . 8 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ x e. A ) -> ( -. ( F ` x ) = 0 -> ( F ` x ) e. RR+ ) )
103	102	ralimdva 2962	. . . . . . 7 |- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( A. x e. A -. ( F ` x ) = 0 -> A. x e. A ( F ` x ) e. RR+ ) )
104	103	imp 445	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> A. x e. A ( F ` x ) e. RR+ )
105		ffnfv 6388	. . . . . 6 |- ( F : A --> RR+ <-> ( F Fn A /\ A. x e. A ( F ` x ) e. RR+ ) )
106	84, 104, 105	sylanbrc 698	. . . . 5 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> F : A --> RR+ )
107	1, 80, 81, 106	amgmlem 24716	. . . 4 |- ( ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) /\ A. x e. A -. ( F ` x ) = 0 ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )
108	107	ex 450	. . 3 |- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( A. x e. A -. ( F ` x ) = 0 -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
109	79, 108	syl5bir 233	. 2 |- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( -. E. x e. A ( F ` x ) = 0 -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) ) )
110	78, 109	pm2.61d 170	1 |- ( ( A e. Fin /\ A =/= (/) /\ F : A --> ( 0 [,) +oo ) ) -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   RR+crp 11832   [,)cico 12177   #chash 13117   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  ℂ_fldccnfld 19746   ^c ccxp 24302

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-inf2 8538  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by: (None)