Theorem amgmlemALT 42549

Description: Alternate proof of amgmlem 24716 using amgmwlem 42548. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Kunhao Zheng, 20-Jun-2021.)

Hypotheses

Ref	Expression
amgmlemALT.0	|- M = (mulGrp ` ℂfld )
amgmlemALT.1	|- ( ph -> A e. Fin )
amgmlemALT.2	|- ( ph -> A =/= (/) )
amgmlemALT.3	|- ( ph -> F : A --> RR+ )

Assertion

Ref	Expression
amgmlemALT	|- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

Proof of Theorem amgmlemALT

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

Step	Hyp	Ref	Expression
1		amgmlemALT.0	. . 3 |- M = (mulGrp ` ℂfld )
2		amgmlemALT.1	. . 3 |- ( ph -> A e. Fin )
3		amgmlemALT.2	. . 3 |- ( ph -> A =/= (/) )
4		amgmlemALT.3	. . 3 |- ( ph -> F : A --> RR+ )
5		hashnncl 13157	. . . . . . . 8 |- ( A e. Fin -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
6	2, 5	syl 17	. . . . . . 7 |- ( ph -> ( ( # ` A ) e. NN <-> A =/= (/) ) )
7	3, 6	mpbird 247	. . . . . 6 |- ( ph -> ( # ` A ) e. NN )
8	7	nnrpd 11870	. . . . 5 |- ( ph -> ( # ` A ) e. RR+ )
9	8	rpreccld 11882	. . . 4 |- ( ph -> ( 1 / ( # ` A ) ) e. RR+ )
10		fconst6g 6094	. . . 4 |- ( ( 1 / ( # ` A ) ) e. RR+ -> ( A X. { ( 1 / ( # ` A ) ) } ) : A --> RR+ )
11	9, 10	syl 17	. . 3 |- ( ph -> ( A X. { ( 1 / ( # ` A ) ) } ) : A --> RR+ )
12		fconstmpt 5163	. . . . . 6 |- ( A X. { ( 1 / ( # ` A ) ) } ) = ( k e. A |-> ( 1 / ( # ` A ) ) )
13	12	a1i 11	. . . . 5 |- ( ph -> ( A X. { ( 1 / ( # ` A ) ) } ) = ( k e. A |-> ( 1 / ( # ` A ) ) ) )
14	13	oveq2d 6666	. . . 4 |- ( ph -> (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) = (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) )
15	7	nnrecred 11066	. . . . . 6 |- ( ph -> ( 1 / ( # ` A ) ) e. RR )
16	15	recnd 10068	. . . . 5 |- ( ph -> ( 1 / ( # ` A ) ) e. CC )
17		simpl 473	. . . . . 6 |- ( ( A e. Fin /\ ( 1 / ( # ` A ) ) e. CC ) -> A e. Fin )
18		simplr 792	. . . . . 6 |- ( ( ( A e. Fin /\ ( 1 / ( # ` A ) ) e. CC ) /\ k e. A ) -> ( 1 / ( # ` A ) ) e. CC )
19	17, 18	gsumfsum 19813	. . . . 5 |- ( ( A e. Fin /\ ( 1 / ( # ` A ) ) e. CC ) -> (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) = sum_ k e. A ( 1 / ( # ` A ) ) )
20	2, 16, 19	syl2anc 693	. . . 4 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( 1 / ( # ` A ) ) ) ) = sum_ k e. A ( 1 / ( # ` A ) ) )
21		fsumconst 14522	. . . . . 6 |- ( ( A e. Fin /\ ( 1 / ( # ` A ) ) e. CC ) -> sum_ k e. A ( 1 / ( # ` A ) ) = ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) )
22	2, 16, 21	syl2anc 693	. . . . 5 |- ( ph -> sum_ k e. A ( 1 / ( # ` A ) ) = ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) )
23	7	nncnd 11036	. . . . . 6 |- ( ph -> ( # ` A ) e. CC )
24	7	nnne0d 11065	. . . . . 6 |- ( ph -> ( # ` A ) =/= 0 )
25	23, 24	recidd 10796	. . . . 5 |- ( ph -> ( ( # ` A ) x. ( 1 / ( # ` A ) ) ) = 1 )
26	22, 25	eqtrd 2656	. . . 4 |- ( ph -> sum_ k e. A ( 1 / ( # ` A ) ) = 1 )
27	14, 20, 26	3eqtrd 2660	. . 3 |- ( ph -> (ℂfld gsumg ( A X. { ( 1 / ( # ` A ) ) } ) ) = 1 )
28	1, 2, 3, 4, 11, 27	amgmwlem 42548	. 2 |- ( ph -> ( M gsumg ( F oF ^c ( A X. { ( 1 / ( # ` A ) ) } ) ) ) <_ (ℂfld gsumg ( F oF x. ( A X. { ( 1 / ( # ` A ) ) } ) ) ) )
29		rpssre 11843	. . . . . 6 |- RR+ C_ RR
30		ax-resscn 9993	. . . . . 6 |- RR C_ CC
31	29, 30	sstri 3612	. . . . 5 |- RR+ C_ CC
32		eqid 2622	. . . . . 6 |- ( M ↾s RR+ ) = ( M ↾s RR+ )
33		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
34	1, 33	mgpbas 18495	. . . . . 6 |- CC = ( Base ` M )
35	32, 34	ressbas2 15931	. . . . 5 |- ( RR+ C_ CC -> RR+ = ( Base ` ( M ↾s RR+ ) ) )
36	31, 35	ax-mp 5	. . . 4 |- RR+ = ( Base ` ( M ↾s RR+ ) )
37		cnfld1 19771	. . . . . 6 |- 1 = ( 1r ` ℂfld )
38	1, 37	ringidval 18503	. . . . 5 |- 1 = ( 0g ` M )
39	1	oveq1i 6660	. . . . . . . . . 10 |- ( M ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
40	39	rpmsubg 19810	. . . . . . . . 9 |- RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) )
41		subgsubm 17616	. . . . . . . . 9 |- ( RR+ e. (SubGrp ` ( M ↾s ( CC \ { 0 } ) ) ) -> RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) )
42	40, 41	ax-mp 5	. . . . . . . 8 |- RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) )
43		cnring 19768	. . . . . . . . . 10 |- ℂfld e. Ring
44		cnfld0 19770	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
45		cndrng 19775	. . . . . . . . . . . 12 |- ℂfld e. DivRing
46	33, 44, 45	drngui 18753	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
47	46, 1	unitsubm 18670	. . . . . . . . . 10 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` M ) )
48	43, 47	ax-mp 5	. . . . . . . . 9 |- ( CC \ { 0 } ) e. (SubMnd ` M )
49		eqid 2622	. . . . . . . . . 10 |- ( M ↾s ( CC \ { 0 } ) ) = ( M ↾s ( CC \ { 0 } ) )
50	49	subsubm 17357	. . . . . . . . 9 |- ( ( CC \ { 0 } ) e. (SubMnd ` M ) -> ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) ) )
51	48, 50	ax-mp 5	. . . . . . . 8 |- ( RR+ e. (SubMnd ` ( M ↾s ( CC \ { 0 } ) ) ) <-> ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) ) )
52	42, 51	mpbi 220	. . . . . . 7 |- ( RR+ e. (SubMnd ` M ) /\ RR+ C_ ( CC \ { 0 } ) )
53	52	simpli 474	. . . . . 6 |- RR+ e. (SubMnd ` M )
54		eqid 2622	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
55	32, 54	subm0 17356	. . . . . 6 |- ( RR+ e. (SubMnd ` M ) -> ( 0g ` M ) = ( 0g ` ( M ↾s RR+ ) ) )
56	53, 55	ax-mp 5	. . . . 5 |- ( 0g ` M ) = ( 0g ` ( M ↾s RR+ ) )
57	38, 56	eqtri 2644	. . . 4 |- 1 = ( 0g ` ( M ↾s RR+ ) )
58		cncrng 19767	. . . . . 6 |- ℂfld e. CRing
59	1	crngmgp 18555	. . . . . 6 |- (ℂfld e. CRing -> M e. CMnd )
60	58, 59	ax-mp 5	. . . . 5 |- M e. CMnd
61	32	submmnd 17354	. . . . . 6 |- ( RR+ e. (SubMnd ` M ) -> ( M ↾s RR+ ) e. Mnd )
62	53, 61	mp1i 13	. . . . 5 |- ( ph -> ( M ↾s RR+ ) e. Mnd )
63	32	subcmn 18242	. . . . 5 |- ( ( M e. CMnd /\ ( M ↾s RR+ ) e. Mnd ) -> ( M ↾s RR+ ) e. CMnd )
64	60, 62, 63	sylancr 695	. . . 4 |- ( ph -> ( M ↾s RR+ ) e. CMnd )
65		reex 10027	. . . . . . . 8 |- RR e. _V
66	65, 29	ssexi 4803	. . . . . . 7 |- RR+ e. _V
67		cnfldmul 19752	. . . . . . . . 9 |- x. = ( .r ` ℂfld )
68	1, 67	mgpplusg 18493	. . . . . . . 8 |- x. = ( +g ` M )
69	32, 68	ressplusg 15993	. . . . . . 7 |- ( RR+ e. _V -> x. = ( +g ` ( M ↾s RR+ ) ) )
70	66, 69	ax-mp 5	. . . . . 6 |- x. = ( +g ` ( M ↾s RR+ ) )
71		eqid 2622	. . . . . . . 8 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
72	71	rpmsubg 19810	. . . . . . 7 |- RR+ e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
73	1	oveq1i 6660	. . . . . . . . 9 |- ( M ↾s RR+ ) = ( (mulGrp ` ℂfld ) ↾s RR+ )
74		cnex 10017	. . . . . . . . . . 11 |- CC e. _V
75		difss 3737	. . . . . . . . . . 11 |- ( CC \ { 0 } ) C_ CC
76	74, 75	ssexi 4803	. . . . . . . . . 10 |- ( CC \ { 0 } ) e. _V
77		rpcndif0 11851	. . . . . . . . . . 11 |- ( w e. RR+ -> w e. ( CC \ { 0 } ) )
78	77	ssriv 3607	. . . . . . . . . 10 |- RR+ C_ ( CC \ { 0 } )
79		ressabs 15939	. . . . . . . . . 10 |- ( ( ( CC \ { 0 } ) e. _V /\ RR+ C_ ( CC \ { 0 } ) ) -> ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s RR+ ) = ( (mulGrp ` ℂfld ) ↾s RR+ ) )
80	76, 78, 79	mp2an 708	. . . . . . . . 9 |- ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s RR+ ) = ( (mulGrp ` ℂfld ) ↾s RR+ )
81	73, 80	eqtr4i 2647	. . . . . . . 8 |- ( M ↾s RR+ ) = ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s RR+ )
82	81	subggrp 17597	. . . . . . 7 |- ( RR+ e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) -> ( M ↾s RR+ ) e. Grp )
83	72, 82	mp1i 13	. . . . . 6 |- ( ph -> ( M ↾s RR+ ) e. Grp )
84		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. RR+ ) -> k e. RR+ )
85	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. RR+ ) -> ( 1 / ( # ` A ) ) e. RR )
86	84, 85	rpcxpcld 24476	. . . . . . 7 |- ( ( ph /\ k e. RR+ ) -> ( k ^c ( 1 / ( # ` A ) ) ) e. RR+ )
87		eqid 2622	. . . . . . 7 |- ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) = ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) )
88	86, 87	fmptd 6385	. . . . . 6 |- ( ph -> ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) : RR+ --> RR+ )
89		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> x e. RR+ )
90	89	rprege0d 11879	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x e. RR /\ 0 <_ x ) )
91		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> y e. RR+ )
92	91	rprege0d 11879	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( y e. RR /\ 0 <_ y ) )
93	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( 1 / ( # ` A ) ) e. CC )
94		mulcxp 24431	. . . . . . . 8 |- ( ( ( x e. RR /\ 0 <_ x ) /\ ( y e. RR /\ 0 <_ y ) /\ ( 1 / ( # ` A ) ) e. CC ) -> ( ( x x. y ) ^c ( 1 / ( # ` A ) ) ) = ( ( x ^c ( 1 / ( # ` A ) ) ) x. ( y ^c ( 1 / ( # ` A ) ) ) ) )
95	90, 92, 93, 94	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( x x. y ) ^c ( 1 / ( # ` A ) ) ) = ( ( x ^c ( 1 / ( # ` A ) ) ) x. ( y ^c ( 1 / ( # ` A ) ) ) ) )
96		rpmulcl 11855	. . . . . . . . 9 |- ( ( x e. RR+ /\ y e. RR+ ) -> ( x x. y ) e. RR+ )
97	96	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( x x. y ) e. RR+ )
98		oveq1 6657	. . . . . . . . 9 |- ( k = ( x x. y ) -> ( k ^c ( 1 / ( # ` A ) ) ) = ( ( x x. y ) ^c ( 1 / ( # ` A ) ) ) )
99		ovex 6678	. . . . . . . . 9 |- ( k ^c ( 1 / ( # ` A ) ) ) e. _V
100	98, 87, 99	fvmpt3i 6287	. . . . . . . 8 |- ( ( x x. y ) e. RR+ -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( x x. y ) ) = ( ( x x. y ) ^c ( 1 / ( # ` A ) ) ) )
101	97, 100	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( x x. y ) ) = ( ( x x. y ) ^c ( 1 / ( # ` A ) ) ) )
102		oveq1 6657	. . . . . . . . . 10 |- ( k = x -> ( k ^c ( 1 / ( # ` A ) ) ) = ( x ^c ( 1 / ( # ` A ) ) ) )
103	102, 87, 99	fvmpt3i 6287	. . . . . . . . 9 |- ( x e. RR+ -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` x ) = ( x ^c ( 1 / ( # ` A ) ) ) )
104	89, 103	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` x ) = ( x ^c ( 1 / ( # ` A ) ) ) )
105		oveq1 6657	. . . . . . . . . 10 |- ( k = y -> ( k ^c ( 1 / ( # ` A ) ) ) = ( y ^c ( 1 / ( # ` A ) ) ) )
106	105, 87, 99	fvmpt3i 6287	. . . . . . . . 9 |- ( y e. RR+ -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` y ) = ( y ^c ( 1 / ( # ` A ) ) ) )
107	91, 106	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` y ) = ( y ^c ( 1 / ( # ` A ) ) ) )
108	104, 107	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` x ) x. ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` y ) ) = ( ( x ^c ( 1 / ( # ` A ) ) ) x. ( y ^c ( 1 / ( # ` A ) ) ) ) )
109	95, 101, 108	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( x e. RR+ /\ y e. RR+ ) ) -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( x x. y ) ) = ( ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` x ) x. ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` y ) ) )
110	36, 36, 70, 70, 83, 83, 88, 109	isghmd 17669	. . . . 5 |- ( ph -> ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) e. ( ( M ↾s RR+ ) GrpHom ( M ↾s RR+ ) ) )
111		ghmmhm 17670	. . . . 5 |- ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) e. ( ( M ↾s RR+ ) GrpHom ( M ↾s RR+ ) ) -> ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) e. ( ( M ↾s RR+ ) MndHom ( M ↾s RR+ ) ) )
112	110, 111	syl 17	. . . 4 |- ( ph -> ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) e. ( ( M ↾s RR+ ) MndHom ( M ↾s RR+ ) ) )
113		1red 10055	. . . . 5 |- ( ph -> 1 e. RR )
114	4, 2, 113	fdmfifsupp 8285	. . . 4 |- ( ph -> F finSupp 1 )
115	36, 57, 64, 62, 2, 112, 4, 114	gsummhm 18338	. . 3 |- ( ph -> ( ( M ↾s RR+ ) gsumg ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) o. F ) ) = ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( ( M ↾s RR+ ) gsumg F ) ) )
116	53	a1i 11	. . . . 5 |- ( ph -> RR+ e. (SubMnd ` M ) )
117	4	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. RR+ )
118	15	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. A ) -> ( 1 / ( # ` A ) ) e. RR )
119	117, 118	rpcxpcld 24476	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) e. RR+ )
120		eqid 2622	. . . . . 6 |- ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) = ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) )
121	119, 120	fmptd 6385	. . . . 5 |- ( ph -> ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) : A --> RR+ )
122	2, 116, 121, 32	gsumsubm 17373	. . . 4 |- ( ph -> ( M gsumg ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) ) = ( ( M ↾s RR+ ) gsumg ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) ) )
123	9	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( 1 / ( # ` A ) ) e. RR+ )
124	4	feqmptd 6249	. . . . . 6 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
125	2, 117, 123, 124, 13	offval2 6914	. . . . 5 |- ( ph -> ( F oF ^c ( A X. { ( 1 / ( # ` A ) ) } ) ) = ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) )
126	125	oveq2d 6666	. . . 4 |- ( ph -> ( M gsumg ( F oF ^c ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( M gsumg ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) ) )
127	102	cbvmptv 4750	. . . . . . 7 |- ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) = ( x e. RR+ |-> ( x ^c ( 1 / ( # ` A ) ) ) )
128	127	a1i 11	. . . . . 6 |- ( ph -> ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) = ( x e. RR+ |-> ( x ^c ( 1 / ( # ` A ) ) ) ) )
129		oveq1 6657	. . . . . 6 |- ( x = ( F ` k ) -> ( x ^c ( 1 / ( # ` A ) ) ) = ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) )
130	117, 124, 128, 129	fmptco 6396	. . . . 5 |- ( ph -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) o. F ) = ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) )
131	130	oveq2d 6666	. . . 4 |- ( ph -> ( ( M ↾s RR+ ) gsumg ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) o. F ) ) = ( ( M ↾s RR+ ) gsumg ( k e. A |-> ( ( F ` k ) ^c ( 1 / ( # ` A ) ) ) ) ) )
132	122, 126, 131	3eqtr4rd 2667	. . 3 |- ( ph -> ( ( M ↾s RR+ ) gsumg ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) o. F ) ) = ( M gsumg ( F oF ^c ( A X. { ( 1 / ( # ` A ) ) } ) ) ) )
133	36, 57, 64, 2, 4, 114	gsumcl 18316	. . . . 5 |- ( ph -> ( ( M ↾s RR+ ) gsumg F ) e. RR+ )
134		oveq1 6657	. . . . . 6 |- ( k = ( ( M ↾s RR+ ) gsumg F ) -> ( k ^c ( 1 / ( # ` A ) ) ) = ( ( ( M ↾s RR+ ) gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
135	134, 87, 99	fvmpt3i 6287	. . . . 5 |- ( ( ( M ↾s RR+ ) gsumg F ) e. RR+ -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( ( M ↾s RR+ ) gsumg F ) ) = ( ( ( M ↾s RR+ ) gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
136	133, 135	syl 17	. . . 4 |- ( ph -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( ( M ↾s RR+ ) gsumg F ) ) = ( ( ( M ↾s RR+ ) gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
137	2, 116, 4, 32	gsumsubm 17373	. . . . 5 |- ( ph -> ( M gsumg F ) = ( ( M ↾s RR+ ) gsumg F ) )
138	137	oveq1d 6665	. . . 4 |- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) = ( ( ( M ↾s RR+ ) gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
139	136, 138	eqtr4d 2659	. . 3 |- ( ph -> ( ( k e. RR+ |-> ( k ^c ( 1 / ( # ` A ) ) ) ) ` ( ( M ↾s RR+ ) gsumg F ) ) = ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
140	115, 132, 139	3eqtr3d 2664	. 2 |- ( ph -> ( M gsumg ( F oF ^c ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) )
141	117	rpcnd 11874	. . . . . . 7 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. CC )
142	2, 141	fsumcl 14464	. . . . . 6 |- ( ph -> sum_ k e. A ( F ` k ) e. CC )
143	142, 23, 24	divrecd 10804	. . . . 5 |- ( ph -> ( sum_ k e. A ( F ` k ) / ( # ` A ) ) = ( sum_ k e. A ( F ` k ) x. ( 1 / ( # ` A ) ) ) )
144	2, 16, 141	fsummulc1 14517	. . . . 5 |- ( ph -> ( sum_ k e. A ( F ` k ) x. ( 1 / ( # ` A ) ) ) = sum_ k e. A ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) )
145	143, 144	eqtr2d 2657	. . . 4 |- ( ph -> sum_ k e. A ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) = ( sum_ k e. A ( F ` k ) / ( # ` A ) ) )
146	16	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( 1 / ( # ` A ) ) e. CC )
147	141, 146	mulcld 10060	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) e. CC )
148	2, 147	gsumfsum 19813	. . . 4 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) ) ) = sum_ k e. A ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) )
149	2, 141	gsumfsum 19813	. . . . 5 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) = sum_ k e. A ( F ` k ) )
150	149	oveq1d 6665	. . . 4 |- ( ph -> ( (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) / ( # ` A ) ) = ( sum_ k e. A ( F ` k ) / ( # ` A ) ) )
151	145, 148, 150	3eqtr4d 2666	. . 3 |- ( ph -> (ℂfld gsumg ( k e. A |-> ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) ) ) = ( (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) / ( # ` A ) ) )
152	2, 117, 146, 124, 13	offval2 6914	. . . 4 |- ( ph -> ( F oF x. ( A X. { ( 1 / ( # ` A ) ) } ) ) = ( k e. A |-> ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) ) )
153	152	oveq2d 6666	. . 3 |- ( ph -> (ℂfld gsumg ( F oF x. ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = (ℂfld gsumg ( k e. A |-> ( ( F ` k ) x. ( 1 / ( # ` A ) ) ) ) ) )
154	124	oveq2d 6666	. . . 4 |- ( ph -> (ℂfld gsumg F ) = (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) )
155	154	oveq1d 6665	. . 3 |- ( ph -> ( (ℂfld gsumg F ) / ( # ` A ) ) = ( (ℂfld gsumg ( k e. A |-> ( F ` k ) ) ) / ( # ` A ) ) )
156	151, 153, 155	3eqtr4d 2666	. 2 |- ( ph -> (ℂfld gsumg ( F oF x. ( A X. { ( 1 / ( # ` A ) ) } ) ) ) = ( (ℂfld gsumg F ) / ( # ` A ) ) )
157	28, 140, 156	3brtr3d 4684	1 |- ( ph -> ( ( M gsumg F ) ^c ( 1 / ( # ` A ) ) ) <_ ( (ℂfld gsumg F ) / ( # ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   / cdiv 10684   NNcn 11020   RR+crp 11832   #chash 13117   sum_csu 14416   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333  SubMndcsubmnd 17334   Grpcgrp 17422  SubGrpcsubg 17588   GrpHom cghm 17657  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)