Theorem psgnunilem5 16645

Description: Lemma for psgnuni 16650. 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 3797	. . . 4 |- -. A e. (/)
2		psgnunilem2.id	. . . . . . . 8 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3	2	difeq1d 3617	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) \ _I ) = ( ( _I |` D ) \ _I ) )
4	3	dmeqd 5215	. . . . . 6 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = dom ( ( _I |` D ) \ _I ) )
5		resss 5307	. . . . . . . . 9 |- ( _I |` D ) C_ _I
6		ssdif0 3888	. . . . . . . . 9 |- ( ( _I |` D ) C_ _I <-> ( ( _I |` D ) \ _I ) = (/) )
7	5, 6	mpbi 208	. . . . . . . 8 |- ( ( _I |` D ) \ _I ) = (/)
8	7	dmeqi 5214	. . . . . . 7 |- dom ( ( _I |` D ) \ _I ) = dom (/)
9		dm0 5226	. . . . . . 7 |- dom (/) = (/)
10	8, 9	eqtri 2486	. . . . . 6 |- dom ( ( _I |` D ) \ _I ) = (/)
11	4, 10	syl6eq 2514	. . . . 5 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = (/) )
12	11	eleq2d 2527	. . . 4 |- ( ph -> ( A e. dom ( ( G gsumg W ) \ _I ) <-> A e. (/) ) )
13	1, 12	mtbiri 303	. . 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 16551	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
17		grpmnd 16188	. . . . . . . . 9 |- ( G e. Grp -> G e. Mnd )
18	14, 16, 17	3syl 20	. . . . . . . 8 |- ( ph -> G e. Mnd )
19		psgnunilem2.t	. . . . . . . . . . . 12 |- T = ran (pmTrsp ` D )
20		eqid 2457	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
21	19, 15, 20	symgtrf 16620	. . . . . . . . . . 11 |- T C_ ( Base ` G )
22		sswrd 12560	. . . . . . . . . . 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 3500	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
26		swrdcl 12654	. . . . . . . . 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 16134	. . . . . . . 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 661	. . . . . . 7 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) )
30	15, 20	symgbasf1o 16534	. . . . . . 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 16	. . . . . 6 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
32	31	adantr 465	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
33		wrdf 12557	. . . . . . . . . 10 |- ( W e. Word T -> W : ( 0..^ ( # ` W ) ) --> T )
34	24, 33	syl 16	. . . . . . . . 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 6312	. . . . . . . . . 10 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ L ) )
38	35, 37	eleqtrrd 2548	. . . . . . . . 9 |- ( ph -> I e. ( 0..^ ( # ` W ) ) )
39	34, 38	ffvelrnd 6033	. . . . . . . 8 |- ( ph -> ( W ` I ) e. T )
40	21, 39	sseldi 3497	. . . . . . 7 |- ( ph -> ( W ` I ) e. ( Base ` G ) )
41	15, 20	symgbasf1o 16534	. . . . . . 7 |- ( ( W ` I ) e. ( Base ` G ) -> ( W ` I ) : D -1-1-onto-> D )
42	40, 41	syl 16	. . . . . 6 |- ( ph -> ( W ` I ) : D -1-1-onto-> D )
43	42	adantr 465	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W ` I ) : D -1-1-onto-> D )
44	15, 20	symgsssg 16618	. . . . . . . . . . . 12 |- ( D e. V -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) )
45		subgsubm 16349	. . . . . . . . . . . 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 20	. . . . . . . . . . 11 |- ( ph -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
47	46	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
48		fzossfz 11843	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ L ) C_ ( 0 ... L )
49	48, 35	sseldi 3497	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> I e. ( 0 ... L ) )
50		elfzuz3 11710	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. ( 0 ... L ) -> L e. ( ZZ>= ` I ) )
51	49, 50	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ( ZZ>= ` I ) )
52	36, 51	eqeltrd 2545	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` W ) e. ( ZZ>= ` I ) )
53		fzoss2 11851	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` W ) e. ( ZZ>= ` I ) -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
54	52, 53	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0..^ I ) C_ ( 0..^ ( # ` W ) ) )
55	54	sselda 3499	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ I ) ) -> s e. ( 0..^ ( # ` W ) ) )
56	34	ffvelrnda 6032	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. T )
57	21, 56	sseldi 3497	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. ( Base ` G ) )
58	55, 57	syldan 470	. . . . . . . . . . . . . . 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 5872	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = s -> ( W ` k ) = ( W ` s ) )
61	60	difeq1d 3617	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = s -> ( ( W ` k ) \ _I ) = ( ( W ` s ) \ _I ) )
62	61	dmeqd 5215	. . . . . . . . . . . . . . . . . . . 20 |- ( k = s -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` s ) \ _I ) )
63	62	eleq2d 2527	. . . . . . . . . . . . . . . . . . 19 |- ( k = s -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` s ) \ _I ) ) )
64	63	notbid 294	. . . . . . . . . . . . . . . . . 18 |- ( k = s -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
65	64	cbvralv 3084	. . . . . . . . . . . . . . . . 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 196	. . . . . . . . . . . . . . . 16 |- ( ph -> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
67	66	r19.21bi 2826	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> -. A e. dom ( ( W ` s ) \ _I ) )
68		difeq1 3611	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( W ` s ) -> ( j \ _I ) = ( ( W ` s ) \ _I ) )
69	68	dmeqd 5215	. . . . . . . . . . . . . . . . . 18 |- ( j = ( W ` s ) -> dom ( j \ _I ) = dom ( ( W ` s ) \ _I ) )
70	69	sseq1d 3526	. . . . . . . . . . . . . . . . 17 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) ) )
71		disj2 3877	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) )
72		disjsn 4092	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( W ` s ) \ _I ) )
73	71, 72	bitr3i 251	. . . . . . . . . . . . . . . . 17 |- ( dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) )
74	70, 73	syl6bb 261	. . . . . . . . . . . . . . . 16 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
75	74	elrab 3257	. . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
77		eqid 2457	. . . . . . . . . . . . . 14 |- ( s e. ( 0..^ I ) |-> ( W ` s ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) )
78	76, 77	fmptd 6056	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
79	36	oveq2d 6312	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) )
80	49, 79	eleqtrrd 2548	. . . . . . . . . . . . . . . 16 |- ( ph -> I e. ( 0 ... ( # ` W ) ) )
81		swrd0val 12656	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word T /\ I e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
82	24, 80, 81	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ph -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
83	34	feqmptd 5926	. . . . . . . . . . . . . . . 16 |- ( ph -> W = ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) )
84	83	reseq1d 5282	. . . . . . . . . . . . . . 15 |- ( ph -> ( W |` ( 0..^ I ) ) = ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) )
85		resmpt 5333	. . . . . . . . . . . . . . . 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 20	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
87	82, 84, 86	3eqtrd 2502	. . . . . . . . . . . . . 14 |- ( ph -> ( W substr <. 0 , I >. ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
88	87	feq1d 5723	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 |- ( ph -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
90	89	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
91		iswrdi 12556	. . . . . . . . . . 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 16	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) e. Word { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
93		gsumwsubmcl 16132	. . . . . . . . . 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 661	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
95		difeq1 3611	. . . . . . . . . . . . . 14 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( j \ _I ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
96	95	dmeqd 5215	. . . . . . . . . . . . 13 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> dom ( j \ _I ) = dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
97	96	sseq1d 3526	. . . . . . . . . . . 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 3257	. . . . . . . . . . 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 464	. . . . . . . . . 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 3877	. . . . . . . . . . 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 4092	. . . . . . . . . . 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 251	. . . . . . . . . 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 196	. . . . . . . . 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 16	. . . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( W ` I ) \ _I ) )
107	104, 106	jca 532	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) )
108	107	olcd 393	. . . . . 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 1368	. . . . . 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 212	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) )
111		f1omvdco3 16600	. . . . 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 1228	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
113	24	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W e. Word T )
114		elfzo0 11861	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) <-> ( I e. NN0 /\ L e. NN /\ I < L ) )
115	114	simp2bi 1012	. . . . . . . . . . . . . 14 |- ( I e. ( 0..^ L ) -> L e. NN )
116	35, 115	syl 16	. . . . . . . . . . . . 13 |- ( ph -> L e. NN )
117	36, 116	eqeltrd 2545	. . . . . . . . . . . 12 |- ( ph -> ( # ` W ) e. NN )
118		wrdfin 12567	. . . . . . . . . . . . 13 |- ( W e. Word T -> W e. Fin )
119		hashnncl 12438	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
120	24, 118, 119	3syl 20	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
121	117, 120	mpbid 210	. . . . . . . . . . 11 |- ( ph -> W =/= (/) )
122	121	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W =/= (/) )
123		swrdccatwrd 12704	. . . . . . . . . . 11 |- ( ( W e. Word T /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = W )
124	123	eqcomd 2465	. . . . . . . . . 10 |- ( ( W e. Word T /\ W =/= (/) ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
125	113, 122, 124	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
126	36	oveq1d 6311	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
127	126	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
128	116	nncnd 10572	. . . . . . . . . . . . 13 |- ( ph -> L e. CC )
129		1cnd 9629	. . . . . . . . . . . . 13 |- ( ph -> 1 e. CC )
130		elfzoelz 11825	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) -> I e. ZZ )
131	35, 130	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> I e. ZZ )
132	131	zcnd 10991	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
133	128, 129, 132	subadd2d 9969	. . . . . . . . . . . 12 |- ( ph -> ( ( L - 1 ) = I <-> ( I + 1 ) = L ) )
134	133	biimpar 485	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( L - 1 ) = I )
135	127, 134	eqtrd 2498	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = I )
136		opeq2 4220	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> <. 0 , ( ( # ` W ) - 1 ) >. = <. 0 , I >. )
137	136	oveq2d 6312	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
138	137	adantl 466	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
139		lsw 12592	. . . . . . . . . . . . . 14 |- ( W e. Word T -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
140	24, 139	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
141		fveq2 5872	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` I ) )
142	140, 141	sylan9eq 2518	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( lastS ` W ) = ( W ` I ) )
143	142	s1eqd 12621	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> <" ( lastS ` W ) "> = <" ( W ` I ) "> )
144	138, 143	oveq12d 6314	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
145	135, 144	syldan 470	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
146	125, 145	eqtrd 2498	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
147	146	oveq2d 6312	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) )
148	40	s1cld 12623	. . . . . . . . 9 |- ( ph -> <" ( W ` I ) "> e. Word ( Base ` G ) )
149		eqid 2457	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
150	20, 149	gsumccat 16135	. . . . . . . . 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 1228	. . . . . . . 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 465	. . . . . . 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 16133	. . . . . . . . . . 11 |- ( ( W ` I ) e. ( Base ` G ) -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
154	40, 153	syl 16	. . . . . . . . . 10 |- ( ph -> ( G gsumg <" ( W ` I ) "> ) = ( W ` I ) )
155	154	oveq2d 6312	. . . . . . . . 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 16541	. . . . . . . . . 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 661	. . . . . . . . 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 2498	. . . . . . . 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 465	. . . . . . 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 2502	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
161	160	difeq1d 3617	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg W ) \ _I ) = ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
162	161	dmeqd 5215	. . . 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 2548	. . 3 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( G gsumg W ) \ _I ) )
164	13, 163	mtand 659	. 2 |- ( ph -> -. ( I + 1 ) = L )
165		fzostep1 11924	. . . 4 |- ( I e. ( 0..^ L ) -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
166	35, 165	syl 16	. . 3 |- ( ph -> ( ( I + 1 ) e. ( 0..^ L ) \/ ( I + 1 ) = L ) )
167	166	ord 377	. 2 |- ( ph -> ( -. ( I + 1 ) e. ( 0..^ L ) -> ( I + 1 ) = L ) )
168	164, 167	mt3d 125	1 |- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/_ wxo 1363   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   \ cdif 3468   i^i cin 3470   C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   class class class wbr 4456   |-> cmpt 4515   _I cid 4799   dom cdm 5008   ran crn 5009   |` cres 5010   o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Fincfn 7535   0cc0 9509   1c1 9510   + caddc 9512   < clt 9645   - cmin 9824   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11820   #chash 12407  Word cword 12537   lastS clsw 12538   ++ cconcat 12539   <"cs1 12540   substr csubstr 12541   Basecbs 14643   +g cplusg 14711   gsum_g cgsu 14857   Mndcmnd 16045  SubMndcsubmnd 16091   Grpcgrp 16179  SubGrpcsubg 16321   SymGrpcsymg 16528  pmTrspcpmtr 16592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1364  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-word 12545  df-lsw 12546  df-concat 12547  df-s1 12548  df-substr 12549  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-tset 14730  df-0g 14858  df-gsum 14859  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-subg 16324  df-symg 16529  df-pmtr 16593

This theorem is referenced by:  psgnunilem2  16646