Theorem psgnunilem5 17122

Description: Lemma for psgnuni 17127. 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 3765	. . . 4 |- -. A e. (/)
2		psgnunilem2.id	. . . . . . . 8 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3	2	difeq1d 3582	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) \ _I ) = ( ( _I |` D ) \ _I ) )
4	3	dmeqd 5052	. . . . . 6 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = dom ( ( _I |` D ) \ _I ) )
5		resss 5143	. . . . . . . . 9 |- ( _I |` D ) C_ _I
6		ssdif0 3851	. . . . . . . . 9 |- ( ( _I |` D ) C_ _I <-> ( ( _I |` D ) \ _I ) = (/) )
7	5, 6	mpbi 211	. . . . . . . 8 |- ( ( _I |` D ) \ _I ) = (/)
8	7	dmeqi 5051	. . . . . . 7 |- dom ( ( _I |` D ) \ _I ) = dom (/)
9		dm0 5063	. . . . . . 7 |- dom (/) = (/)
10	8, 9	eqtri 2451	. . . . . 6 |- dom ( ( _I |` D ) \ _I ) = (/)
11	4, 10	syl6eq 2479	. . . . 5 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = (/) )
12	11	eleq2d 2492	. . . 4 |- ( ph -> ( A e. dom ( ( G gsumg W ) \ _I ) <-> A e. (/) ) )
13	1, 12	mtbiri 304	. . 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 17028	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
17		grpmnd 16665	. . . . . . . . 9 |- ( G e. Grp -> G e. Mnd )
18	14, 16, 17	3syl 18	. . . . . . . 8 |- ( ph -> G e. Mnd )
19		psgnunilem2.t	. . . . . . . . . . . 12 |- T = ran (pmTrsp ` D )
20		eqid 2422	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
21	19, 15, 20	symgtrf 17097	. . . . . . . . . . 11 |- T C_ ( Base ` G )
22		sswrd 12671	. . . . . . . . . . 11 |- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) )
23	21, 22	mp1i 13	. . . . . . . . . 10 |- ( ph -> Word T C_ Word ( Base ` G ) )
24		psgnunilem2.w	. . . . . . . . . 10 |- ( ph -> W e. Word T )
25	23, 24	sseldd 3465	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
26		swrdcl 12765	. . . . . . . . 9 |- ( W e. Word ( Base ` G ) -> ( W substr <. 0 , I >. ) e. Word ( Base ` G ) )
27	25, 26	syl 17	. . . . . . . 8 |- ( ph -> ( W substr <. 0 , I >. ) e. Word ( Base ` G ) )
28	20	gsumwcl 16611	. . . . . . . 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 665	. . . . . . 7 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) )
30	15, 20	symgbasf1o 17011	. . . . . . 7 |- ( ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
31	29, 30	syl 17	. . . . . 6 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
32	31	adantr 466	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
33		wrdf 12668	. . . . . . . . . 10 |- ( W e. Word T -> W : ( 0..^ ( # ` W ) ) --> T )
34	24, 33	syl 17	. . . . . . . . 9 |- ( ph -> W : ( 0..^ ( # ` W ) ) --> T )
35		psgnunilem2.ix	. . . . . . . . . 10 |- ( ph -> I e. ( 0..^ L ) )
36		psgnunilem2.l	. . . . . . . . . . 11 |- ( ph -> ( # ` W ) = L )
37	36	oveq2d 6317	. . . . . . . . . 10 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ L ) )
38	35, 37	eleqtrrd 2513	. . . . . . . . 9 |- ( ph -> I e. ( 0..^ ( # ` W ) ) )
39	34, 38	ffvelrnd 6034	. . . . . . . 8 |- ( ph -> ( W ` I ) e. T )
40	21, 39	sseldi 3462	. . . . . . 7 |- ( ph -> ( W ` I ) e. ( Base ` G ) )
41	15, 20	symgbasf1o 17011	. . . . . . 7 |- ( ( W ` I ) e. ( Base ` G ) -> ( W ` I ) : D -1-1-onto-> D )
42	40, 41	syl 17	. . . . . 6 |- ( ph -> ( W ` I ) : D -1-1-onto-> D )
43	42	adantr 466	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W ` I ) : D -1-1-onto-> D )
44	15, 20	symgsssg 17095	. . . . . . . . . . . 12 |- ( D e. V -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) )
45		subgsubm 16826	. . . . . . . . . . . 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 ) )
46	14, 44, 45	3syl 18	. . . . . . . . . . 11 |- ( ph -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
47	46	adantr 466	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
48		fzossfz 11938	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ L ) C_ ( 0 ... L )
49	48, 35	sseldi 3462	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> I e. ( 0 ... L ) )
50		elfzuz3 11797	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. ( 0 ... L ) -> L e. ( ZZ>= ` I ) )
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ( ZZ>= ` I ) )
52	36, 51	eqeltrd 2510	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` W ) e. ( ZZ>= ` I ) )
53		fzoss2 11946	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` W ) e. ( ZZ>= ` I ) -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
55	54	sselda 3464	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ I ) ) -> s e. ( 0..^ ( # ` W ) ) )
56	34	ffvelrnda 6033	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. T )
57	21, 56	sseldi 3462	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. ( Base ` G ) )
58	55, 57	syldan 472	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. ( Base ` G ) )
59		psgnunilem2.al	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. k e. ( 0..^ I ) -. A e. dom ( ( W ` k ) \ _I ) )
60		fveq2 5877	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = s -> ( W ` k ) = ( W ` s ) )
61	60	difeq1d 3582	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = s -> ( ( W ` k ) \ _I ) = ( ( W ` s ) \ _I ) )
62	61	dmeqd 5052	. . . . . . . . . . . . . . . . . . . 20 |- ( k = s -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` s ) \ _I ) )
63	62	eleq2d 2492	. . . . . . . . . . . . . . . . . . 19 |- ( k = s -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` s ) \ _I ) ) )
64	63	notbid 295	. . . . . . . . . . . . . . . . . 18 |- ( k = s -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
65	64	cbvralv 3055	. . . . . . . . . . . . . . . . 17 |- ( A. k e. ( 0..^ I ) -. A e. dom ( ( W ` k ) \ _I ) <-> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
66	59, 65	sylib 199	. . . . . . . . . . . . . . . 16 |- ( ph -> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
67	66	r19.21bi 2794	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> -. A e. dom ( ( W ` s ) \ _I ) )
68		difeq1 3576	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( W ` s ) -> ( j \ _I ) = ( ( W ` s ) \ _I ) )
69	68	dmeqd 5052	. . . . . . . . . . . . . . . . . 18 |- ( j = ( W ` s ) -> dom ( j \ _I ) = dom ( ( W ` s ) \ _I ) )
70	69	sseq1d 3491	. . . . . . . . . . . . . . . . 17 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) ) )
71		disj2 3840	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) )
72		disjsn 4057	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( W ` s ) \ _I ) )
73	71, 72	bitr3i 254	. . . . . . . . . . . . . . . . 17 |- ( dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) )
74	70, 73	syl6bb 264	. . . . . . . . . . . . . . . 16 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
75	74	elrab 3229	. . . . . . . . . . . . . . 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 ) ) )
76	58, 67, 75	sylanbrc 668	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
77		eqid 2422	. . . . . . . . . . . . . 14 |- ( s e. ( 0..^ I ) |-> ( W ` s ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) )
78	76, 77	fmptd 6057	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
79	36	oveq2d 6317	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) )
80	49, 79	eleqtrrd 2513	. . . . . . . . . . . . . . . 16 |- ( ph -> I e. ( 0 ... ( # ` W ) ) )
81		swrd0val 12767	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word T /\ I e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
82	24, 80, 81	syl2anc 665	. . . . . . . . . . . . . . 15 |- ( ph -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
83	34	feqmptd 5930	. . . . . . . . . . . . . . . 16 |- ( ph -> W = ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) )
84	83	reseq1d 5119	. . . . . . . . . . . . . . 15 |- ( ph -> ( W |` ( 0..^ I ) ) = ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) )
85		resmpt 5169	. . . . . . . . . . . . . . . 16 |- ( ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
86	52, 53, 85	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
87	82, 84, 86	3eqtrd 2467	. . . . . . . . . . . . . 14 |- ( ph -> ( W substr <. 0 , I >. ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
88	87	feq1d 5728	. . . . . . . . . . . . 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 } ) } ) )
89	78, 88	mpbird 235	. . . . . . . . . . . 12 |- ( ph -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
90	89	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
91		iswrdi 12667	. . . . . . . . . . 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 } ) } )
92	90, 91	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) e. Word { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
93		gsumwsubmcl 16609	. . . . . . . . . 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 } ) } )
94	47, 92, 93	syl2anc 665	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
95		difeq1 3576	. . . . . . . . . . . . . 14 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( j \ _I ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
96	95	dmeqd 5052	. . . . . . . . . . . . 13 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> dom ( j \ _I ) = dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
97	96	sseq1d 3491	. . . . . . . . . . . 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 } ) ) )
98	97	elrab 3229	. . . . . . . . . . 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 } ) ) )
99	98	simprbi 465	. . . . . . . . . 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 } ) )
100		disj2 3840	. . . . . . . . . . 11 |- ( ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) i^i { A } ) = (/) <-> dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) )
101		disjsn 4057	. . . . . . . . . . 11 |- ( ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
102	100, 101	bitr3i 254	. . . . . . . . . 10 |- ( dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
103	99, 102	sylib 199	. . . . . . . . 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 ) )
104	94, 103	syl 17	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
105		psgnunilem2.a	. . . . . . . . 9 |- ( ph -> A e. dom ( ( W ` I ) \ _I ) )
106	105	adantr 466	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( W ` I ) \ _I ) )
107	104, 106	jca 534	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) )
108	107	olcd 394	. . . . . 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 ) ) ) )
109		excxor 1406	. . . . . 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 ) ) ) )
110	108, 109	sylibr 215	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) )
111		f1omvdco3 17077	. . . . 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 ) )
112	32, 43, 110, 111	syl3anc 1264	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
113	24	adantr 466	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W e. Word T )
114		elfzo0 11956	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) <-> ( I e. NN0 /\ L e. NN /\ I < L ) )
115	114	simp2bi 1021	. . . . . . . . . . . . . 14 |- ( I e. ( 0..^ L ) -> L e. NN )
116	35, 115	syl 17	. . . . . . . . . . . . 13 |- ( ph -> L e. NN )
117	36, 116	eqeltrd 2510	. . . . . . . . . . . 12 |- ( ph -> ( # ` W ) e. NN )
118		wrdfin 12678	. . . . . . . . . . . . 13 |- ( W e. Word T -> W e. Fin )
119		hashnncl 12546	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
120	24, 118, 119	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
121	117, 120	mpbid 213	. . . . . . . . . . 11 |- ( ph -> W =/= (/) )
122	121	adantr 466	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W =/= (/) )
123		swrdccatwrd 12814	. . . . . . . . . . 11 |- ( ( W e. Word T /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = W )
124	123	eqcomd 2430	. . . . . . . . . 10 |- ( ( W e. Word T /\ W =/= (/) ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
125	113, 122, 124	syl2anc 665	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
126	36	oveq1d 6316	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
127	126	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
128	116	nncnd 10625	. . . . . . . . . . . . 13 |- ( ph -> L e. CC )
129		1cnd 9659	. . . . . . . . . . . . 13 |- ( ph -> 1 e. CC )
130		elfzoelz 11920	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) -> I e. ZZ )
131	35, 130	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> I e. ZZ )
132	131	zcnd 11041	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
133	128, 129, 132	subadd2d 10005	. . . . . . . . . . . 12 |- ( ph -> ( ( L - 1 ) = I <-> ( I + 1 ) = L ) )
134	133	biimpar 487	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( L - 1 ) = I )
135	127, 134	eqtrd 2463	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = I )
136		opeq2 4185	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> <. 0 , ( ( # ` W ) - 1 ) >. = <. 0 , I >. )
137	136	oveq2d 6317	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
138	137	adantl 467	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
139		lsw 12703	. . . . . . . . . . . . . 14 |- ( W e. Word T -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
140	24, 139	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
141		fveq2 5877	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` I ) )
142	140, 141	sylan9eq 2483	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( lastS ` W ) = ( W ` I ) )
143	142	s1eqd 12732	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> <" ( lastS ` W ) "> = <" ( W ` I ) "> )
144	138, 143	oveq12d 6319	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
145	135, 144	syldan 472	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
146	125, 145	eqtrd 2463	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
147	146	oveq2d 6317	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) )
148	40	s1cld 12734	. . . . . . . . 9 |- ( ph -> <" ( W ` I ) "> e. Word ( Base ` G ) )
149		eqid 2422	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
150	20, 149	gsumccat 16612	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( W substr <. 0 , I >. ) e. Word ( Base ` G ) /\ <" ( W ` I ) "> e. Word ( Base ` G ) ) -> ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
151	18, 27, 148, 150	syl3anc 1264	. . . . . . . 8 |- ( ph -> ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
152	151	adantr 466	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) )
153	20	gsumws1 16610	. . . . . . . . . . 11 |- ( ( W ` I ) e. ( Base ` G ) -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
154	40, 153	syl 17	. . . . . . . . . 10 |- ( ph -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
155	154	oveq2d 6317	. . . . . . . . 9 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( W ` I ) ) )
156	15, 20, 149	symgov 17018	. . . . . . . . . 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 ) ) )
157	29, 40, 156	syl2anc 665	. . . . . . . . 9 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( W ` I ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
158	155, 157	eqtrd 2463	. . . . . . . 8 |- ( ph -> ( ( G gsumg ( W substr <. 0 , I >. ) ) ( +g ` G ) ( G gsumg <" ( W ` I ) "> ) ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
159	158	adantr 466	. . . . . . 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 ) ) )
160	147, 152, 159	3eqtrd 2467	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
161	160	difeq1d 3582	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg W ) \ _I ) = ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
162	161	dmeqd 5052	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> dom ( ( G gsumg W ) \ _I ) = dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
163	112, 162	eleqtrrd 2513	. . 3 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( G gsumg W ) \ _I ) )
164	13, 163	mtand 663	. 2 |- ( ph -> -. ( I + 1 ) = L )
165		fzostep1 12020	. . . 4 |- ( I e. ( 0..^ L ) -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
166	35, 165	syl 17	. . 3 |- ( ph -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
167	166	ord 378	. 2 |- ( ph -> ( -. ( I + 1 ) e. ( 0..^ L ) -> ( I + 1 ) = L ) )
168	164, 167	mt3d 128	1 |- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   \/_ wxo 1400   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   {crab 2779   _Vcvv 3081   \ cdif 3433   i^i cin 3435   C_ wss 3436   (/)c0 3761   {csn 3996   <.cop 4002   class class class wbr 4420   |-> cmpt 4479   _I cid 4759   dom cdm 4849   ran crn 4850   |` cres 4851   o. ccom 4853   -->wf 5593   -1-1-onto->wf1o 5596   ` cfv 5597  (class class class)co 6301   Fincfn 7573   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784  ..^cfzo 11915   #chash 12514  Word cword 12648   lastS clsw 12649   ++ cconcat 12650   <"cs1 12651   substr csubstr 12652   Basecbs 15108   +g cplusg 15177   gsum_g cgsu 15326   Mndcmnd 16522  SubMndcsubmnd 16568   Grpcgrp 16656  SubGrpcsubg 16798   SymGrpcsymg 17005  pmTrspcpmtr 17069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12213  df-hash 12515  df-word 12656  df-lsw 12657  df-concat 12658  df-s1 12659  df-substr 12660  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-tset 15196  df-0g 15327  df-gsum 15328  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-grp 16660  df-minusg 16661  df-subg 16801  df-symg 17006  df-pmtr 17070

This theorem is referenced by:  psgnunilem2  17123