Theorem pgpfaclem1 18480

Description: Lemma for pgpfac 18483. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
pgpfac.b	|- B = ( Base ` G )
pgpfac.c	|- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
pgpfac.g	|- ( ph -> G e. Abel )
pgpfac.p	|- ( ph -> P pGrp G )
pgpfac.f	|- ( ph -> B e. Fin )
pgpfac.u	|- ( ph -> U e. (SubGrp ` G ) )
pgpfac.a	|- ( ph -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
pgpfac.h	|- H = ( G ↾s U )
pgpfac.k	|- K = (mrCls ` (SubGrp ` H ) )
pgpfac.o	|- O = ( od ` H )
pgpfac.e	|- E = (gEx ` H )
pgpfac.0	|- .0. = ( 0g ` H )
pgpfac.l	|- .(+) = ( LSSum ` H )
pgpfac.1	|- ( ph -> E =/= 1 )
pgpfac.x	|- ( ph -> X e. U )
pgpfac.oe	|- ( ph -> ( O ` X ) = E )
pgpfac.w	|- ( ph -> W e. (SubGrp ` H ) )
pgpfac.i	|- ( ph -> ( ( K ` { X } ) i^i W ) = { .0. } )
pgpfac.s	|- ( ph -> ( ( K ` { X } ) .(+) W ) = U )
pgpfac.2	|- ( ph -> S e. Word C )
pgpfac.4	|- ( ph -> G dom DProd S )
pgpfac.5	|- ( ph -> ( G DProd S ) = W )
pgpfac.t	|- T = ( S ++ <" ( K ` { X } ) "> )

Assertion

Ref	Expression
pgpfaclem1	|- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Distinct variable groups:   t, s, C   s, r, t, G   K, r, s   ph, t   B, s, t   U, r, s, t   W, s, t   X, r, s   T, s

Allowed substitution hints:   ph( s, r)   B( r)   C( r)   P( t, s, r)   .(+) ( t, s, r)   S( t, s, r)   T( t, r)   E( t, s, r)   H( t, s, r)   K( t)   O( t, s, r)   W( r)   X( t)   .0. ( t, s, r)

Proof of Theorem pgpfaclem1

Step	Hyp	Ref	Expression
1		pgpfac.t	. . 3 |- T = ( S ++ <" ( K ` { X } ) "> )
2		pgpfac.2	. . 3 |- ( ph -> S e. Word C )
3		pgpfac.u	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` G ) )
4		pgpfac.h	. . . . . . . . . 10 |- H = ( G ↾s U )
5	4	subggrp 17597	. . . . . . . . 9 |- ( U e. (SubGrp ` G ) -> H e. Grp )
6	3, 5	syl 17	. . . . . . . 8 |- ( ph -> H e. Grp )
7		eqid 2622	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
8	7	subgacs 17629	. . . . . . . 8 |- ( H e. Grp -> (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) )
9		acsmre 16313	. . . . . . . 8 |- ( (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
10	6, 8, 9	3syl 18	. . . . . . 7 |- ( ph -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
11		pgpfac.x	. . . . . . . 8 |- ( ph -> X e. U )
12	4	subgbas 17598	. . . . . . . . 9 |- ( U e. (SubGrp ` G ) -> U = ( Base ` H ) )
13	3, 12	syl 17	. . . . . . . 8 |- ( ph -> U = ( Base ` H ) )
14	11, 13	eleqtrd 2703	. . . . . . 7 |- ( ph -> X e. ( Base ` H ) )
15		pgpfac.k	. . . . . . . 8 |- K = (mrCls ` (SubGrp ` H ) )
16	15	mrcsncl 16272	. . . . . . 7 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ X e. ( Base ` H ) ) -> ( K ` { X } ) e. (SubGrp ` H ) )
17	10, 14, 16	syl2anc 693	. . . . . 6 |- ( ph -> ( K ` { X } ) e. (SubGrp ` H ) )
18	4	subsubg 17617	. . . . . . 7 |- ( U e. (SubGrp ` G ) -> ( ( K ` { X } ) e. (SubGrp ` H ) <-> ( ( K ` { X } ) e. (SubGrp ` G ) /\ ( K ` { X } ) C_ U ) ) )
19	3, 18	syl 17	. . . . . 6 |- ( ph -> ( ( K ` { X } ) e. (SubGrp ` H ) <-> ( ( K ` { X } ) e. (SubGrp ` G ) /\ ( K ` { X } ) C_ U ) ) )
20	17, 19	mpbid 222	. . . . 5 |- ( ph -> ( ( K ` { X } ) e. (SubGrp ` G ) /\ ( K ` { X } ) C_ U ) )
21	20	simpld 475	. . . 4 |- ( ph -> ( K ` { X } ) e. (SubGrp ` G ) )
22	4	oveq1i 6660	. . . . . . 7 |- ( H ↾s ( K ` { X } ) ) = ( ( G ↾s U ) ↾s ( K ` { X } ) )
23	20	simprd 479	. . . . . . . 8 |- ( ph -> ( K ` { X } ) C_ U )
24		ressabs 15939	. . . . . . . 8 |- ( ( U e. (SubGrp ` G ) /\ ( K ` { X } ) C_ U ) -> ( ( G ↾s U ) ↾s ( K ` { X } ) ) = ( G ↾s ( K ` { X } ) ) )
25	3, 23, 24	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( G ↾s U ) ↾s ( K ` { X } ) ) = ( G ↾s ( K ` { X } ) ) )
26	22, 25	syl5eq 2668	. . . . . 6 |- ( ph -> ( H ↾s ( K ` { X } ) ) = ( G ↾s ( K ` { X } ) ) )
27	7, 15	cycsubgcyg2 18303	. . . . . . 7 |- ( ( H e. Grp /\ X e. ( Base ` H ) ) -> ( H ↾s ( K ` { X } ) ) e. CycGrp )
28	6, 14, 27	syl2anc 693	. . . . . 6 |- ( ph -> ( H ↾s ( K ` { X } ) ) e. CycGrp )
29	26, 28	eqeltrrd 2702	. . . . 5 |- ( ph -> ( G ↾s ( K ` { X } ) ) e. CycGrp )
30		pgpfac.p	. . . . . . 7 |- ( ph -> P pGrp G )
31		pgpprm 18008	. . . . . . 7 |- ( P pGrp G -> P e. Prime )
32	30, 31	syl 17	. . . . . 6 |- ( ph -> P e. Prime )
33		subgpgp 18012	. . . . . . 7 |- ( ( P pGrp G /\ ( K ` { X } ) e. (SubGrp ` G ) ) -> P pGrp ( G ↾s ( K ` { X } ) ) )
34	30, 21, 33	syl2anc 693	. . . . . 6 |- ( ph -> P pGrp ( G ↾s ( K ` { X } ) ) )
35		brelrng 5355	. . . . . 6 |- ( ( P e. Prime /\ ( G ↾s ( K ` { X } ) ) e. CycGrp /\ P pGrp ( G ↾s ( K ` { X } ) ) ) -> ( G ↾s ( K ` { X } ) ) e. ran pGrp )
36	32, 29, 34, 35	syl3anc 1326	. . . . 5 |- ( ph -> ( G ↾s ( K ` { X } ) ) e. ran pGrp )
37	29, 36	elind 3798	. . . 4 |- ( ph -> ( G ↾s ( K ` { X } ) ) e. (CycGrp i^i ran pGrp ) )
38		oveq2 6658	. . . . . 6 |- ( r = ( K ` { X } ) -> ( G ↾s r ) = ( G ↾s ( K ` { X } ) ) )
39	38	eleq1d 2686	. . . . 5 |- ( r = ( K ` { X } ) -> ( ( G ↾s r ) e. (CycGrp i^i ran pGrp ) <-> ( G ↾s ( K ` { X } ) ) e. (CycGrp i^i ran pGrp ) ) )
40		pgpfac.c	. . . . 5 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
41	39, 40	elrab2 3366	. . . 4 |- ( ( K ` { X } ) e. C <-> ( ( K ` { X } ) e. (SubGrp ` G ) /\ ( G ↾s ( K ` { X } ) ) e. (CycGrp i^i ran pGrp ) ) )
42	21, 37, 41	sylanbrc 698	. . 3 |- ( ph -> ( K ` { X } ) e. C )
43	1, 2, 42	cats1cld 13600	. 2 |- ( ph -> T e. Word C )
44		wrdf 13310	. . . . 5 |- ( T e. Word C -> T : ( 0..^ ( # ` T ) ) --> C )
45	43, 44	syl 17	. . . 4 |- ( ph -> T : ( 0..^ ( # ` T ) ) --> C )
46		ssrab2 3687	. . . . 5 |- { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) } C_ (SubGrp ` G )
47	40, 46	eqsstri 3635	. . . 4 |- C C_ (SubGrp ` G )
48		fss 6056	. . . 4 |- ( ( T : ( 0..^ ( # ` T ) ) --> C /\ C C_ (SubGrp ` G ) ) -> T : ( 0..^ ( # ` T ) ) --> (SubGrp ` G ) )
49	45, 47, 48	sylancl 694	. . 3 |- ( ph -> T : ( 0..^ ( # ` T ) ) --> (SubGrp ` G ) )
50		fzodisj 12502	. . . 4 |- ( ( 0..^ ( # ` S ) ) i^i ( ( # ` S )..^ ( ( # ` S ) + 1 ) ) ) = (/)
51		lencl 13324	. . . . . . . 8 |- ( S e. Word C -> ( # ` S ) e. NN0 )
52	2, 51	syl 17	. . . . . . 7 |- ( ph -> ( # ` S ) e. NN0 )
53	52	nn0zd 11480	. . . . . 6 |- ( ph -> ( # ` S ) e. ZZ )
54		fzosn 12538	. . . . . 6 |- ( ( # ` S ) e. ZZ -> ( ( # ` S )..^ ( ( # ` S ) + 1 ) ) = { ( # ` S ) } )
55	53, 54	syl 17	. . . . 5 |- ( ph -> ( ( # ` S )..^ ( ( # ` S ) + 1 ) ) = { ( # ` S ) } )
56	55	ineq2d 3814	. . . 4 |- ( ph -> ( ( 0..^ ( # ` S ) ) i^i ( ( # ` S )..^ ( ( # ` S ) + 1 ) ) ) = ( ( 0..^ ( # ` S ) ) i^i { ( # ` S ) } ) )
57	50, 56	syl5reqr 2671	. . 3 |- ( ph -> ( ( 0..^ ( # ` S ) ) i^i { ( # ` S ) } ) = (/) )
58	1	fveq2i 6194	. . . . . . 7 |- ( # ` T ) = ( # ` ( S ++ <" ( K ` { X } ) "> ) )
59	42	s1cld 13383	. . . . . . . 8 |- ( ph -> <" ( K ` { X } ) "> e. Word C )
60		ccatlen 13360	. . . . . . . 8 |- ( ( S e. Word C /\ <" ( K ` { X } ) "> e. Word C ) -> ( # ` ( S ++ <" ( K ` { X } ) "> ) ) = ( ( # ` S ) + ( # ` <" ( K ` { X } ) "> ) ) )
61	2, 59, 60	syl2anc 693	. . . . . . 7 |- ( ph -> ( # ` ( S ++ <" ( K ` { X } ) "> ) ) = ( ( # ` S ) + ( # ` <" ( K ` { X } ) "> ) ) )
62	58, 61	syl5eq 2668	. . . . . 6 |- ( ph -> ( # ` T ) = ( ( # ` S ) + ( # ` <" ( K ` { X } ) "> ) ) )
63		s1len 13385	. . . . . . 7 |- ( # ` <" ( K ` { X } ) "> ) = 1
64	63	oveq2i 6661	. . . . . 6 |- ( ( # ` S ) + ( # ` <" ( K ` { X } ) "> ) ) = ( ( # ` S ) + 1 )
65	62, 64	syl6eq 2672	. . . . 5 |- ( ph -> ( # ` T ) = ( ( # ` S ) + 1 ) )
66	65	oveq2d 6666	. . . 4 |- ( ph -> ( 0..^ ( # ` T ) ) = ( 0..^ ( ( # ` S ) + 1 ) ) )
67		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
68	52, 67	syl6eleq 2711	. . . . 5 |- ( ph -> ( # ` S ) e. ( ZZ>= ` 0 ) )
69		fzosplitsn 12576	. . . . 5 |- ( ( # ` S ) e. ( ZZ>= ` 0 ) -> ( 0..^ ( ( # ` S ) + 1 ) ) = ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } ) )
70	68, 69	syl 17	. . . 4 |- ( ph -> ( 0..^ ( ( # ` S ) + 1 ) ) = ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } ) )
71	66, 70	eqtrd 2656	. . 3 |- ( ph -> ( 0..^ ( # ` T ) ) = ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } ) )
72		eqid 2622	. . 3 |- (Cntz ` G ) = (Cntz ` G )
73		eqid 2622	. . 3 |- ( 0g ` G ) = ( 0g ` G )
74		pgpfac.4	. . . 4 |- ( ph -> G dom DProd S )
75		cats1un 13475	. . . . . . . 8 |- ( ( S e. Word C /\ ( K ` { X } ) e. C ) -> ( S ++ <" ( K ` { X } ) "> ) = ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) )
76	2, 42, 75	syl2anc 693	. . . . . . 7 |- ( ph -> ( S ++ <" ( K ` { X } ) "> ) = ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) )
77	1, 76	syl5eq 2668	. . . . . 6 |- ( ph -> T = ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) )
78	77	reseq1d 5395	. . . . 5 |- ( ph -> ( T |` ( 0..^ ( # ` S ) ) ) = ( ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) |` ( 0..^ ( # ` S ) ) ) )
79		wrdf 13310	. . . . . . 7 |- ( S e. Word C -> S : ( 0..^ ( # ` S ) ) --> C )
80		ffn 6045	. . . . . . 7 |- ( S : ( 0..^ ( # ` S ) ) --> C -> S Fn ( 0..^ ( # ` S ) ) )
81	2, 79, 80	3syl 18	. . . . . 6 |- ( ph -> S Fn ( 0..^ ( # ` S ) ) )
82		fzonel 12483	. . . . . 6 |- -. ( # ` S ) e. ( 0..^ ( # ` S ) )
83		fsnunres 6454	. . . . . 6 |- ( ( S Fn ( 0..^ ( # ` S ) ) /\ -. ( # ` S ) e. ( 0..^ ( # ` S ) ) ) -> ( ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) |` ( 0..^ ( # ` S ) ) ) = S )
84	81, 82, 83	sylancl 694	. . . . 5 |- ( ph -> ( ( S u. { <. ( # ` S ) , ( K ` { X } ) >. } ) |` ( 0..^ ( # ` S ) ) ) = S )
85	78, 84	eqtrd 2656	. . . 4 |- ( ph -> ( T |` ( 0..^ ( # ` S ) ) ) = S )
86	74, 85	breqtrrd 4681	. . 3 |- ( ph -> G dom DProd ( T |` ( 0..^ ( # ` S ) ) ) )
87		fvex 6201	. . . . . 6 |- ( # ` S ) e. _V
88		dprdsn 18435	. . . . . 6 |- ( ( ( # ` S ) e. _V /\ ( K ` { X } ) e. (SubGrp ` G ) ) -> ( G dom DProd { <. ( # ` S ) , ( K ` { X } ) >. } /\ ( G DProd { <. ( # ` S ) , ( K ` { X } ) >. } ) = ( K ` { X } ) ) )
89	87, 21, 88	sylancr 695	. . . . 5 |- ( ph -> ( G dom DProd { <. ( # ` S ) , ( K ` { X } ) >. } /\ ( G DProd { <. ( # ` S ) , ( K ` { X } ) >. } ) = ( K ` { X } ) ) )
90	89	simpld 475	. . . 4 |- ( ph -> G dom DProd { <. ( # ` S ) , ( K ` { X } ) >. } )
91		ffn 6045	. . . . . . 7 |- ( T : ( 0..^ ( # ` T ) ) --> C -> T Fn ( 0..^ ( # ` T ) ) )
92	43, 44, 91	3syl 18	. . . . . 6 |- ( ph -> T Fn ( 0..^ ( # ` T ) ) )
93		ssun2 3777	. . . . . . . 8 |- { ( # ` S ) } C_ ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } )
94	87	snss 4316	. . . . . . . 8 |- ( ( # ` S ) e. ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } ) <-> { ( # ` S ) } C_ ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } ) )
95	93, 94	mpbir 221	. . . . . . 7 |- ( # ` S ) e. ( ( 0..^ ( # ` S ) ) u. { ( # ` S ) } )
96	95, 71	syl5eleqr 2708	. . . . . 6 |- ( ph -> ( # ` S ) e. ( 0..^ ( # ` T ) ) )
97		fnressn 6425	. . . . . 6 |- ( ( T Fn ( 0..^ ( # ` T ) ) /\ ( # ` S ) e. ( 0..^ ( # ` T ) ) ) -> ( T |` { ( # ` S ) } ) = { <. ( # ` S ) , ( T ` ( # ` S ) ) >. } )
98	92, 96, 97	syl2anc 693	. . . . 5 |- ( ph -> ( T |` { ( # ` S ) } ) = { <. ( # ` S ) , ( T ` ( # ` S ) ) >. } )
99	1	fveq1i 6192	. . . . . . . . 9 |- ( T ` ( # ` S ) ) = ( ( S ++ <" ( K ` { X } ) "> ) ` ( # ` S ) )
100	52	nn0cnd 11353	. . . . . . . . . . . 12 |- ( ph -> ( # ` S ) e. CC )
101	100	addid2d 10237	. . . . . . . . . . 11 |- ( ph -> ( 0 + ( # ` S ) ) = ( # ` S ) )
102	101	eqcomd 2628	. . . . . . . . . 10 |- ( ph -> ( # ` S ) = ( 0 + ( # ` S ) ) )
103	102	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( ( S ++ <" ( K ` { X } ) "> ) ` ( # ` S ) ) = ( ( S ++ <" ( K ` { X } ) "> ) ` ( 0 + ( # ` S ) ) ) )
104	99, 103	syl5eq 2668	. . . . . . . 8 |- ( ph -> ( T ` ( # ` S ) ) = ( ( S ++ <" ( K ` { X } ) "> ) ` ( 0 + ( # ` S ) ) ) )
105		1nn 11031	. . . . . . . . . . . 12 |- 1 e. NN
106	105	a1i 11	. . . . . . . . . . 11 |- ( ph -> 1 e. NN )
107	63, 106	syl5eqel 2705	. . . . . . . . . 10 |- ( ph -> ( # ` <" ( K ` { X } ) "> ) e. NN )
108		lbfzo0 12507	. . . . . . . . . 10 |- ( 0 e. ( 0..^ ( # ` <" ( K ` { X } ) "> ) ) <-> ( # ` <" ( K ` { X } ) "> ) e. NN )
109	107, 108	sylibr 224	. . . . . . . . 9 |- ( ph -> 0 e. ( 0..^ ( # ` <" ( K ` { X } ) "> ) ) )
110		ccatval3 13363	. . . . . . . . 9 |- ( ( S e. Word C /\ <" ( K ` { X } ) "> e. Word C /\ 0 e. ( 0..^ ( # ` <" ( K ` { X } ) "> ) ) ) -> ( ( S ++ <" ( K ` { X } ) "> ) ` ( 0 + ( # ` S ) ) ) = ( <" ( K ` { X } ) "> ` 0 ) )
111	2, 59, 109, 110	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( ( S ++ <" ( K ` { X } ) "> ) ` ( 0 + ( # ` S ) ) ) = ( <" ( K ` { X } ) "> ` 0 ) )
112		fvex 6201	. . . . . . . . 9 |- ( K ` { X } ) e. _V
113		s1fv 13390	. . . . . . . . 9 |- ( ( K ` { X } ) e. _V -> ( <" ( K ` { X } ) "> ` 0 ) = ( K ` { X } ) )
114	112, 113	mp1i 13	. . . . . . . 8 |- ( ph -> ( <" ( K ` { X } ) "> ` 0 ) = ( K ` { X } ) )
115	104, 111, 114	3eqtrd 2660	. . . . . . 7 |- ( ph -> ( T ` ( # ` S ) ) = ( K ` { X } ) )
116	115	opeq2d 4409	. . . . . 6 |- ( ph -> <. ( # ` S ) , ( T ` ( # ` S ) ) >. = <. ( # ` S ) , ( K ` { X } ) >. )
117	116	sneqd 4189	. . . . 5 |- ( ph -> { <. ( # ` S ) , ( T ` ( # ` S ) ) >. } = { <. ( # ` S ) , ( K ` { X } ) >. } )
118	98, 117	eqtrd 2656	. . . 4 |- ( ph -> ( T |` { ( # ` S ) } ) = { <. ( # ` S ) , ( K ` { X } ) >. } )
119	90, 118	breqtrrd 4681	. . 3 |- ( ph -> G dom DProd ( T |` { ( # ` S ) } ) )
120		pgpfac.g	. . . 4 |- ( ph -> G e. Abel )
121		dprdsubg 18423	. . . . 5 |- ( G dom DProd ( T |` ( 0..^ ( # ` S ) ) ) -> ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) e. (SubGrp ` G ) )
122	86, 121	syl 17	. . . 4 |- ( ph -> ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) e. (SubGrp ` G ) )
123		dprdsubg 18423	. . . . 5 |- ( G dom DProd ( T |` { ( # ` S ) } ) -> ( G DProd ( T |` { ( # ` S ) } ) ) e. (SubGrp ` G ) )
124	119, 123	syl 17	. . . 4 |- ( ph -> ( G DProd ( T |` { ( # ` S ) } ) ) e. (SubGrp ` G ) )
125	72, 120, 122, 124	ablcntzd 18260	. . 3 |- ( ph -> ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) C_ ( (Cntz ` G ) ` ( G DProd ( T |` { ( # ` S ) } ) ) ) )
126		pgpfac.i	. . . 4 |- ( ph -> ( ( K ` { X } ) i^i W ) = { .0. } )
127	85	oveq2d 6666	. . . . . . 7 |- ( ph -> ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) = ( G DProd S ) )
128		pgpfac.5	. . . . . . 7 |- ( ph -> ( G DProd S ) = W )
129	127, 128	eqtrd 2656	. . . . . 6 |- ( ph -> ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) = W )
130	118	oveq2d 6666	. . . . . . 7 |- ( ph -> ( G DProd ( T |` { ( # ` S ) } ) ) = ( G DProd { <. ( # ` S ) , ( K ` { X } ) >. } ) )
131	89	simprd 479	. . . . . . 7 |- ( ph -> ( G DProd { <. ( # ` S ) , ( K ` { X } ) >. } ) = ( K ` { X } ) )
132	130, 131	eqtrd 2656	. . . . . 6 |- ( ph -> ( G DProd ( T |` { ( # ` S ) } ) ) = ( K ` { X } ) )
133	129, 132	ineq12d 3815	. . . . 5 |- ( ph -> ( ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) i^i ( G DProd ( T |` { ( # ` S ) } ) ) ) = ( W i^i ( K ` { X } ) ) )
134		incom 3805	. . . . 5 |- ( W i^i ( K ` { X } ) ) = ( ( K ` { X } ) i^i W )
135	133, 134	syl6eq 2672	. . . 4 |- ( ph -> ( ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) i^i ( G DProd ( T |` { ( # ` S ) } ) ) ) = ( ( K ` { X } ) i^i W ) )
136	4, 73	subg0 17600	. . . . . . 7 |- ( U e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
137	3, 136	syl 17	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
138		pgpfac.0	. . . . . 6 |- .0. = ( 0g ` H )
139	137, 138	syl6eqr 2674	. . . . 5 |- ( ph -> ( 0g ` G ) = .0. )
140	139	sneqd 4189	. . . 4 |- ( ph -> { ( 0g ` G ) } = { .0. } )
141	126, 135, 140	3eqtr4d 2666	. . 3 |- ( ph -> ( ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) i^i ( G DProd ( T |` { ( # ` S ) } ) ) ) = { ( 0g ` G ) } )
142	49, 57, 71, 72, 73, 86, 119, 125, 141	dmdprdsplit2 18445	. 2 |- ( ph -> G dom DProd T )
143		eqid 2622	. . . . 5 |- ( LSSum ` G ) = ( LSSum ` G )
144	49, 57, 71, 143, 142	dprdsplit 18447	. . . 4 |- ( ph -> ( G DProd T ) = ( ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) ( LSSum ` G ) ( G DProd ( T |` { ( # ` S ) } ) ) ) )
145	129, 132	oveq12d 6668	. . . 4 |- ( ph -> ( ( G DProd ( T |` ( 0..^ ( # ` S ) ) ) ) ( LSSum ` G ) ( G DProd ( T |` { ( # ` S ) } ) ) ) = ( W ( LSSum ` G ) ( K ` { X } ) ) )
146	129, 122	eqeltrrd 2702	. . . . 5 |- ( ph -> W e. (SubGrp ` G ) )
147	143	lsmcom 18261	. . . . 5 |- ( ( G e. Abel /\ W e. (SubGrp ` G ) /\ ( K ` { X } ) e. (SubGrp ` G ) ) -> ( W ( LSSum ` G ) ( K ` { X } ) ) = ( ( K ` { X } ) ( LSSum ` G ) W ) )
148	120, 146, 21, 147	syl3anc 1326	. . . 4 |- ( ph -> ( W ( LSSum ` G ) ( K ` { X } ) ) = ( ( K ` { X } ) ( LSSum ` G ) W ) )
149	144, 145, 148	3eqtrd 2660	. . 3 |- ( ph -> ( G DProd T ) = ( ( K ` { X } ) ( LSSum ` G ) W ) )
150		pgpfac.w	. . . . . 6 |- ( ph -> W e. (SubGrp ` H ) )
151	7	subgss 17595	. . . . . 6 |- ( W e. (SubGrp ` H ) -> W C_ ( Base ` H ) )
152	150, 151	syl 17	. . . . 5 |- ( ph -> W C_ ( Base ` H ) )
153	152, 13	sseqtr4d 3642	. . . 4 |- ( ph -> W C_ U )
154		pgpfac.l	. . . . 5 |- .(+) = ( LSSum ` H )
155	4, 143, 154	subglsm 18086	. . . 4 |- ( ( U e. (SubGrp ` G ) /\ ( K ` { X } ) C_ U /\ W C_ U ) -> ( ( K ` { X } ) ( LSSum ` G ) W ) = ( ( K ` { X } ) .(+) W ) )
156	3, 23, 153, 155	syl3anc 1326	. . 3 |- ( ph -> ( ( K ` { X } ) ( LSSum ` G ) W ) = ( ( K ` { X } ) .(+) W ) )
157		pgpfac.s	. . 3 |- ( ph -> ( ( K ` { X } ) .(+) W ) = U )
158	149, 156, 157	3eqtrd 2660	. 2 |- ( ph -> ( G DProd T ) = U )
159		breq2 4657	. . . 4 |- ( s = T -> ( G dom DProd s <-> G dom DProd T ) )
160		oveq2 6658	. . . . 5 |- ( s = T -> ( G DProd s ) = ( G DProd T ) )
161	160	eqeq1d 2624	. . . 4 |- ( s = T -> ( ( G DProd s ) = U <-> ( G DProd T ) = U ) )
162	159, 161	anbi12d 747	. . 3 |- ( s = T -> ( ( G dom DProd s /\ ( G DProd s ) = U ) <-> ( G dom DProd T /\ ( G DProd T ) = U ) ) )
163	162	rspcev 3309	. 2 |- ( ( T e. Word C /\ ( G dom DProd T /\ ( G DProd T ) = U ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
164	43, 142, 158, 163	syl12anc 1324	1 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   Primecprime 15385   Basecbs 15857   ↾_s cress 15858   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   odcod 17944  gExcgex 17945   pGrp cpgp 17946   LSSumclsm 18049   Abelcabl 18194  CycGrpccyg 18279   DProd cdprd 18392

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

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-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-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  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-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-od 17948  df-pgp 17950  df-lsm 18051  df-cmn 18195  df-abl 18196  df-cyg 18280  df-dprd 18394

This theorem is referenced by:  pgpfaclem2  18481