Theorem psgnunilem5 27285

Description: Lemma for psgnuni 27290. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
psgnunilem2.g	|- G = ( SymGrp ` D )
psgnunilem2.t	|- T = ran (pmTrsp ` D )
psgnunilem2.d	|- ( ph -> D e. V )
psgnunilem2.w	|- ( ph -> W e. Word T )
psgnunilem2.id	|- ( ph -> ( G gsumg W ) = ( _I |` D ) )
psgnunilem2.l	|- ( ph -> ( # ` W ) = L )
psgnunilem2.ix	|- ( ph -> I e. ( 0..^ L ) )
psgnunilem2.a	|- ( ph -> A e. dom ( ( W ` I ) \ _I ) )
psgnunilem2.al	|- ( ph -> A. k e. ( 0..^ I ) -. A e. dom ( ( W ` k ) \ _I ) )

Assertion

Ref	Expression
psgnunilem5	|- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Distinct variable groups:   A, k   k, G   k, I   k, W

Allowed substitution hints:   ph( k)   D( k)   T( k)   L( k)   V( k)

Proof of Theorem psgnunilem5

Dummy variables j s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3592	. . . 4 |- -. A e. (/)
2		psgnunilem2.id	. . . . . . . 8 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3	2	difeq1d 3424	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) \ _I ) = ( ( _I |` D ) \ _I ) )
4	3	dmeqd 5031	. . . . . 6 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = dom ( ( _I |` D ) \ _I ) )
5		resss 5129	. . . . . . . . 9 |- ( _I |` D ) C_ _I
6		ssdif0 3646	. . . . . . . . 9 |- ( ( _I |` D ) C_ _I <-> ( ( _I |` D ) \ _I ) = (/) )
7	5, 6	mpbi 200	. . . . . . . 8 |- ( ( _I |` D ) \ _I ) = (/)
8	7	dmeqi 5030	. . . . . . 7 |- dom ( ( _I |` D ) \ _I ) = dom (/)
9		dm0 5042	. . . . . . 7 |- dom (/) = (/)
10	8, 9	eqtri 2424	. . . . . 6 |- dom ( ( _I |` D ) \ _I ) = (/)
11	4, 10	syl6eq 2452	. . . . 5 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = (/) )
12	11	eleq2d 2471	. . . 4 |- ( ph -> ( A e. dom ( ( G gsumg W ) \ _I ) <-> A e. (/) ) )
13	1, 12	mtbiri 295	. . 3 |- ( ph -> -. A e. dom ( ( G gsumg W ) \ _I ) )
14		psgnunilem2.d	. . . . . . . . 9 |- ( ph -> D e. V )
15		psgnunilem2.g	. . . . . . . . . 10 |- G = ( SymGrp ` D )
16	15	symggrp 15058	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
17		grpmnd 14772	. . . . . . . . 9 |- ( G e. Grp -> G e. Mnd )
18	14, 16, 17	3syl 19	. . . . . . . 8 |- ( ph -> G e. Mnd )
19		psgnunilem2.t	. . . . . . . . . . . 12 |- T = ran (pmTrsp ` D )
20		eqid 2404	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
21	19, 15, 20	symgtrf 27278	. . . . . . . . . . 11 |- T C_ ( Base ` G )
22		sswrd 11692	. . . . . . . . . . 11 |- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) )
23	21, 22	mp1i 12	. . . . . . . . . 10 |- ( ph -> Word T C_ Word ( Base ` G ) )
24		psgnunilem2.w	. . . . . . . . . 10 |- ( ph -> W e. Word T )
25	23, 24	sseldd 3309	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
26		swrdcl 11721	. . . . . . . . 9 |- ( W e. Word ( Base ` G ) -> ( W substr <. 0 , I >. ) e. Word ( Base ` G ) )
27	25, 26	syl 16	. . . . . . . 8 |- ( ph -> ( W substr <. 0 , I >. ) e. Word ( Base ` G ) )
28	20	gsumwcl 14741	. . . . . . . 8 |- ( ( G e. Mnd /\ ( W substr <. 0 , I >. ) e. Word ( Base ` G ) ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) )
29	18, 27, 28	syl2anc 643	. . . . . . 7 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) )
30	15, 20	elsymgbas2 15051	. . . . . . . 8 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) -> ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) <-> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D ) )
31	30	ibi 233	. . . . . . 7 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
32	29, 31	syl 16	. . . . . 6 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
33	32	adantr 452	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
34		wrdf 11688	. . . . . . . . . 10 |- ( W e. Word T -> W : ( 0..^ ( # ` W ) ) --> T )
35	24, 34	syl 16	. . . . . . . . 9 |- ( ph -> W : ( 0..^ ( # ` W ) ) --> T )
36		psgnunilem2.ix	. . . . . . . . . 10 |- ( ph -> I e. ( 0..^ L ) )
37		psgnunilem2.l	. . . . . . . . . . 11 |- ( ph -> ( # ` W ) = L )
38	37	oveq2d 6056	. . . . . . . . . 10 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ L ) )
39	36, 38	eleqtrrd 2481	. . . . . . . . 9 |- ( ph -> I e. ( 0..^ ( # ` W ) ) )
40	35, 39	ffvelrnd 5830	. . . . . . . 8 |- ( ph -> ( W ` I ) e. T )
41	21, 40	sseldi 3306	. . . . . . 7 |- ( ph -> ( W ` I ) e. ( Base ` G ) )
42	15, 20	elsymgbas2 15051	. . . . . . . 8 |- ( ( W ` I ) e. ( Base ` G ) -> ( ( W ` I ) e. ( Base ` G ) <-> ( W ` I ) : D -1-1-onto-> D ) )
43	42	ibi 233	. . . . . . 7 |- ( ( W ` I ) e. ( Base ` G ) -> ( W ` I ) : D -1-1-onto-> D )
44	41, 43	syl 16	. . . . . 6 |- ( ph -> ( W ` I ) : D -1-1-onto-> D )
45	44	adantr 452	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W ` I ) : D -1-1-onto-> D )
46	15, 20	symgsssg 27276	. . . . . . . . . . . 12 |- ( D e. V -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) )
47		subgsubm 14917	. . . . . . . . . . . 12 |- ( { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
48	14, 46, 47	3syl 19	. . . . . . . . . . 11 |- ( ph -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
49	48	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
50		fzossfz 11112	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ L ) C_ ( 0 ... L )
51	50, 36	sseldi 3306	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> I e. ( 0 ... L ) )
52		elfzuz3 11012	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. ( 0 ... L ) -> L e. ( ZZ>= ` I ) )
53	51, 52	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ( ZZ>= ` I ) )
54	37, 53	eqeltrd 2478	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` W ) e. ( ZZ>= ` I ) )
55		fzoss2 11118	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` W ) e. ( ZZ>= ` I ) -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
56	54, 55	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
57	56	sselda 3308	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ I ) ) -> s e. ( 0..^ ( # ` W ) ) )
58	35	ffvelrnda 5829	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. T )
59	21, 58	sseldi 3306	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. ( Base ` G ) )
60	57, 59	syldan 457	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. ( Base ` G ) )
61		psgnunilem2.al	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. k e. ( 0..^ I ) -. A e. dom ( ( W ` k ) \ _I ) )
62		fveq2 5687	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = s -> ( W ` k ) = ( W ` s ) )
63	62	difeq1d 3424	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = s -> ( ( W ` k ) \ _I ) = ( ( W ` s ) \ _I ) )
64	63	dmeqd 5031	. . . . . . . . . . . . . . . . . . . 20 |- ( k = s -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` s ) \ _I ) )
65	64	eleq2d 2471	. . . . . . . . . . . . . . . . . . 19 |- ( k = s -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` s ) \ _I ) ) )
66	65	notbid 286	. . . . . . . . . . . . . . . . . 18 |- ( k = s -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
67	66	cbvralv 2892	. . . . . . . . . . . . . . . . 17 |- ( A. k e. ( 0..^ I ) -. A e. dom ( ( W ` k ) \ _I ) <-> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
68	61, 67	sylib 189	. . . . . . . . . . . . . . . 16 |- ( ph -> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
69	68	r19.21bi 2764	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> -. A e. dom ( ( W ` s ) \ _I ) )
70		difeq1 3418	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( W ` s ) -> ( j \ _I ) = ( ( W ` s ) \ _I ) )
71	70	dmeqd 5031	. . . . . . . . . . . . . . . . . 18 |- ( j = ( W ` s ) -> dom ( j \ _I ) = dom ( ( W ` s ) \ _I ) )
72	71	sseq1d 3335	. . . . . . . . . . . . . . . . 17 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) ) )
73		disj2 3635	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) )
74		disjsn 3828	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( W ` s ) \ _I ) )
75	73, 74	bitr3i 243	. . . . . . . . . . . . . . . . 17 |- ( dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) )
76	72, 75	syl6bb 253	. . . . . . . . . . . . . . . 16 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
77	76	elrab 3052	. . . . . . . . . . . . . . 15 |- ( ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } <-> ( ( W ` s ) e. ( Base ` G ) /\ -. A e. dom ( ( W ` s ) \ _I ) ) )
78	60, 69, 77	sylanbrc 646	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
79		eqid 2404	. . . . . . . . . . . . . 14 |- ( s e. ( 0..^ I ) |-> ( W ` s ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) )
80	78, 79	fmptd 5852	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
81	37	oveq2d 6056	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) )
82	51, 81	eleqtrrd 2481	. . . . . . . . . . . . . . . 16 |- ( ph -> I e. ( 0 ... ( # ` W ) ) )
83		swrd0val 11723	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word T /\ I e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
84	24, 82, 83	syl2anc 643	. . . . . . . . . . . . . . 15 |- ( ph -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
85	35	feqmptd 5738	. . . . . . . . . . . . . . . 16 |- ( ph -> W = ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) )
86	85	reseq1d 5104	. . . . . . . . . . . . . . 15 |- ( ph -> ( W |` ( 0..^ I ) ) = ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) )
87		resmpt 5150	. . . . . . . . . . . . . . . 16 |- ( ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
88	54, 55, 87	3syl 19	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
89	84, 86, 88	3eqtrd 2440	. . . . . . . . . . . . . 14 |- ( ph -> ( W substr <. 0 , I >. ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
90	89	feq1d 5539	. . . . . . . . . . . . 13 |- ( ph -> ( ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } <-> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } ) )
91	80, 90	mpbird 224	. . . . . . . . . . . 12 |- ( ph -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
92	91	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
93		iswrdi 11686	. . . . . . . . . . 11 |- ( ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } -> ( W substr <. 0 , I >. ) e. Word { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
94	92, 93	syl 16	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) e. Word { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
95		gsumwsubmcl 14739	. . . . . . . . . 10 |- ( ( { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) /\ ( W substr <. 0 , I >. ) e. Word { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
96	49, 94, 95	syl2anc 643	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
97		difeq1 3418	. . . . . . . . . . . . . 14 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( j \ _I ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
98	97	dmeqd 5031	. . . . . . . . . . . . 13 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> dom ( j \ _I ) = dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
99	98	sseq1d 3335	. . . . . . . . . . . 12 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) ) )
100	99	elrab 3052	. . . . . . . . . . 11 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } <-> ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) /\ dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) ) )
101	100	simprbi 451	. . . . . . . . . 10 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } -> dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) )
102		disj2 3635	. . . . . . . . . . 11 |- ( ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) i^i { A } ) = (/) <-> dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) )
103		disjsn 3828	. . . . . . . . . . 11 |- ( ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
104	102, 103	bitr3i 243	. . . . . . . . . 10 |- ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
105	101, 104	sylib 189	. . . . . . . . 9 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } -> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
106	96, 105	syl 16	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
107		psgnunilem2.a	. . . . . . . . 9 |- ( ph -> A e. dom ( ( W ` I ) \ _I ) )
108	107	adantr 452	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( W ` I ) \ _I ) )
109	106, 108	jca 519	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) )
110	109	olcd 383	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ -. A e. dom ( ( W ` I ) \ _I ) ) \/ ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) ) )
111		excxor 1315	. . . . . 6 |- ( ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) <-> ( ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ -. A e. dom ( ( W ` I ) \ _I ) ) \/ ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) ) )
112	110, 111	sylibr 204	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) )
113		f1omvdco3 27260	. . . . 5 |- ( ( ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D /\ ( W ` I ) : D -1-1-onto-> D /\ ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
114	33, 45, 112, 113	syl3anc 1184	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
115	24	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W e. Word T )
116		elfzo0 11126	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) <-> ( I e. NN0 /\ L e. NN /\ I < L ) )
117	116	simp2bi 973	. . . . . . . . . . . . . 14 |- ( I e. ( 0..^ L ) -> L e. NN )
118	36, 117	syl 16	. . . . . . . . . . . . 13 |- ( ph -> L e. NN )
119	37, 118	eqeltrd 2478	. . . . . . . . . . . 12 |- ( ph -> ( # ` W ) e. NN )
120		wrdfin 11689	. . . . . . . . . . . . 13 |- ( W e. Word T -> W e. Fin )
121		hashnncl 11600	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
122	24, 120, 121	3syl 19	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
123	119, 122	mpbid 202	. . . . . . . . . . 11 |- ( ph -> W =/= (/) )
124	123	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W =/= (/) )
125		wrdeqcats1 11743	. . . . . . . . . 10 |- ( ( W e. Word T /\ W =/= (/) ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) )
126	115, 124, 125	syl2anc 643	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) )
127	37	oveq1d 6055	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
128	127	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
129	118	nncnd 9972	. . . . . . . . . . . . 13 |- ( ph -> L e. CC )
130		ax-1cn 9004	. . . . . . . . . . . . . 14 |- 1 e. CC
131	130	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 1 e. CC )
132		elfzoelz 11095	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) -> I e. ZZ )
133	36, 132	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> I e. ZZ )
134	133	zcnd 10332	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
135	129, 131, 134	subadd2d 9386	. . . . . . . . . . . 12 |- ( ph -> ( ( L - 1 ) = I <-> ( I + 1 ) = L ) )
136	135	biimpar 472	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( L - 1 ) = I )
137	128, 136	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = I )
138		opeq2 3945	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> <. 0 , ( ( # ` W ) - 1 ) >. = <. 0 , I >. )
139	138	oveq2d 6056	. . . . . . . . . . 11 |- ( ( ( # ` W ) - 1 ) = I -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
140		fveq2 5687	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` I ) )
141	140	s1eqd 11709	. . . . . . . . . . 11 |- ( ( ( # ` W ) - 1 ) = I -> <" ( W ` ( ( # ` W ) - 1 ) ) "> = <" ( W ` I ) "> )
142	139, 141	oveq12d 6058	. . . . . . . . . 10 |- ( ( ( # ` W ) - 1 ) = I -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) = ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) )
143	137, 142	syl 16	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) = ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) )
144	126, 143	eqtrd 2436	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) )
145	144	oveq2d 6056	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( G gsumg ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) ) )
146	41	s1cld 11711	. . . . . . . . 9 |- ( ph -> <" ( W ` I ) "> e. Word ( Base ` G ) )
147		eqid 2404	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
148	20, 147	gsumccat 14742	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( W substr <. 0 , I >. ) e. Word ( Base ` G ) /\ <" ( W ` I ) "> e. Word ( Base ` G ) ) -> ( G gsumg ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
149	18, 27, 146, 148	syl3anc 1184	. . . . . . . 8 |- ( ph -> ( G gsumg ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
150	149	adantr 452	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
151	20	gsumws1 14740	. . . . . . . . . . 11 |- ( ( W ` I ) e. ( Base ` G ) -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
152	41, 151	syl 16	. . . . . . . . . 10 |- ( ph -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
153	152	oveq2d 6056	. . . . . . . . 9 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( W ` I ) ) )
154	15, 20, 147	symgov 15055	. . . . . . . . . 10 |- ( ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) /\ ( W ` I ) e. ( Base ` G ) ) -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( W ` I ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
155	29, 41, 154	syl2anc 643	. . . . . . . . 9 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( W ` I ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
156	153, 155	eqtrd 2436	. . . . . . . 8 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
157	156	adantr 452	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
158	145, 150, 157	3eqtrd 2440	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
159	158	difeq1d 3424	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg W ) \ _I ) = ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
160	159	dmeqd 5031	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> dom ( ( G gsumg W ) \ _I ) = dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
161	114, 160	eleqtrrd 2481	. . 3 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( G gsumg W ) \ _I ) )
162	13, 161	mtand 641	. 2 |- ( ph -> -. ( I + 1 ) = L )
163		fzostep1 11152	. . . 4 |- ( I e. ( 0..^ L ) -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
164	36, 163	syl 16	. . 3 |- ( ph -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
165	164	ord 367	. 2 |- ( ph -> ( -. ( I + 1 ) e. ( 0..^ L ) -> ( I + 1 ) = L ) )
166	162, 165	mt3d 119	1 |- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   \/_ wxo 1310   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916   \ cdif 3277   i^i cin 3279   C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   e. cmpt 4226   _I cid 4453   dom cdm 4837   ran crn 4838   |` cres 4839   o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947   + caddc 8949   < clt 9076   - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   <"cs1 11674   substr csubstr 11675   Basecbs 13424   +g cplusg 13484   gsum_g cgsu 13679   Mndcmnd 14639   Grpcgrp 14640  SubMndcsubmnd 14692  SubGrpcsubg 14893   SymGrpcsymg 15047  pmTrspcpmtr 27252

This theorem is referenced by:  psgnunilem2  27286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-tset 13503  df-0g 13682  df-gsum 13683  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-symg 15048  df-pmtr 27253