Theorem psgnunilem5 16111

Description: Lemma for psgnuni 16116. 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 3742	. . . 4 |- -. A e. (/)
2		psgnunilem2.id	. . . . . . . 8 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3	2	difeq1d 3574	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) \ _I ) = ( ( _I |` D ) \ _I ) )
4	3	dmeqd 5143	. . . . . 6 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = dom ( ( _I |` D ) \ _I ) )
5		resss 5235	. . . . . . . . 9 |- ( _I |` D ) C_ _I
6		ssdif0 3838	. . . . . . . . 9 |- ( ( _I |` D ) C_ _I <-> ( ( _I |` D ) \ _I ) = (/) )
7	5, 6	mpbi 208	. . . . . . . 8 |- ( ( _I |` D ) \ _I ) = (/)
8	7	dmeqi 5142	. . . . . . 7 |- dom ( ( _I |` D ) \ _I ) = dom (/)
9		dm0 5154	. . . . . . 7 |- dom (/) = (/)
10	8, 9	eqtri 2480	. . . . . 6 |- dom ( ( _I |` D ) \ _I ) = (/)
11	4, 10	syl6eq 2508	. . . . 5 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = (/) )
12	11	eleq2d 2521	. . . 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 16016	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
17		grpmnd 15661	. . . . . . . . 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 2451	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
21	19, 15, 20	symgtrf 16086	. . . . . . . . . . 11 |- T C_ ( Base ` G )
22		sswrd 12353	. . . . . . . . . . 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 3458	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
26		swrdcl 12426	. . . . . . . . 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 15629	. . . . . . . 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	elsymgbas2 15997	. . . . . . . 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 241	. . . . . . 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 465	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
34		wrdf 12351	. . . . . . . . . 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 6209	. . . . . . . . . 10 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ L ) )
39	36, 38	eleqtrrd 2542	. . . . . . . . 9 |- ( ph -> I e. ( 0..^ ( # ` W ) ) )
40	35, 39	ffvelrnd 5946	. . . . . . . 8 |- ( ph -> ( W ` I ) e. T )
41	21, 40	sseldi 3455	. . . . . . 7 |- ( ph -> ( W ` I ) e. ( Base ` G ) )
42	15, 20	elsymgbas2 15997	. . . . . . . 8 |- ( ( W ` I ) e. ( Base ` G ) -> ( ( W ` I ) e. ( Base ` G ) <-> ( W ` I ) : D -1-1-onto-> D ) )
43	42	ibi 241	. . . . . . 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 465	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W ` I ) : D -1-1-onto-> D )
46	15, 20	symgsssg 16084	. . . . . . . . . . . 12 |- ( D e. V -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) )
47		subgsubm 15814	. . . . . . . . . . . 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 20	. . . . . . . . . . 11 |- ( ph -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
49	48	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
50		fzossfz 11680	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ L ) C_ ( 0 ... L )
51	50, 36	sseldi 3455	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> I e. ( 0 ... L ) )
52		elfzuz3 11560	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. ( 0 ... L ) -> L e. ( ZZ>= ` I ) )
53	51, 52	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ( ZZ>= ` I ) )
54	37, 53	eqeltrd 2539	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` W ) e. ( ZZ>= ` I ) )
55		fzoss2 11687	. . . . . . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ I ) ) -> s e. ( 0..^ ( # ` W ) ) )
58	35	ffvelrnda 5945	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. T )
59	21, 58	sseldi 3455	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. ( Base ` G ) )
60	57, 59	syldan 470	. . . . . . . . . . . . . . 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 5792	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = s -> ( W ` k ) = ( W ` s ) )
63	62	difeq1d 3574	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = s -> ( ( W ` k ) \ _I ) = ( ( W ` s ) \ _I ) )
64	63	dmeqd 5143	. . . . . . . . . . . . . . . . . . . 20 |- ( k = s -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` s ) \ _I ) )
65	64	eleq2d 2521	. . . . . . . . . . . . . . . . . . 19 |- ( k = s -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` s ) \ _I ) ) )
66	65	notbid 294	. . . . . . . . . . . . . . . . . 18 |- ( k = s -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
67	66	cbvralv 3046	. . . . . . . . . . . . . . . . 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 196	. . . . . . . . . . . . . . . 16 |- ( ph -> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
69	68	r19.21bi 2913	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> -. A e. dom ( ( W ` s ) \ _I ) )
70		difeq1 3568	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( W ` s ) -> ( j \ _I ) = ( ( W ` s ) \ _I ) )
71	70	dmeqd 5143	. . . . . . . . . . . . . . . . . 18 |- ( j = ( W ` s ) -> dom ( j \ _I ) = dom ( ( W ` s ) \ _I ) )
72	71	sseq1d 3484	. . . . . . . . . . . . . . . . 17 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) ) )
73		disj2 3827	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) )
74		disjsn 4037	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( W ` s ) \ _I ) )
75	73, 74	bitr3i 251	. . . . . . . . . . . . . . . . 17 |- ( dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) )
76	72, 75	syl6bb 261	. . . . . . . . . . . . . . . 16 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
77	76	elrab 3217	. . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
79		eqid 2451	. . . . . . . . . . . . . 14 |- ( s e. ( 0..^ I ) |-> ( W ` s ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) )
80	78, 79	fmptd 5969	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
81	37	oveq2d 6209	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) )
82	51, 81	eleqtrrd 2542	. . . . . . . . . . . . . . . 16 |- ( ph -> I e. ( 0 ... ( # ` W ) ) )
83		swrd0val 12428	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word T /\ I e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
84	24, 82, 83	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ph -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
85	35	feqmptd 5846	. . . . . . . . . . . . . . . 16 |- ( ph -> W = ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) )
86	85	reseq1d 5210	. . . . . . . . . . . . . . 15 |- ( ph -> ( W |` ( 0..^ I ) ) = ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) )
87		resmpt 5257	. . . . . . . . . . . . . . . 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 20	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
89	84, 86, 88	3eqtrd 2496	. . . . . . . . . . . . . 14 |- ( ph -> ( W substr <. 0 , I >. ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
90	89	feq1d 5647	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 |- ( ph -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
92	91	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
93		iswrdi 12350	. . . . . . . . . . 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 15627	. . . . . . . . . 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 661	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
97		difeq1 3568	. . . . . . . . . . . . . 14 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( j \ _I ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
98	97	dmeqd 5143	. . . . . . . . . . . . 13 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> dom ( j \ _I ) = dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
99	98	sseq1d 3484	. . . . . . . . . . . 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 3217	. . . . . . . . . . 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 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 } ) )
102		disj2 3827	. . . . . . . . . . 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 4037	. . . . . . . . . . 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 251	. . . . . . . . . 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 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 ) )
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 465	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( W ` I ) \ _I ) )
109	106, 108	jca 532	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) )
110	109	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 ) ) ) )
111		excxor 1356	. . . . . 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 212	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) )
113		f1omvdco3 16066	. . . . 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 1219	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
115	24	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W e. Word T )
116		elfzo0 11697	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) <-> ( I e. NN0 /\ L e. NN /\ I < L ) )
117	116	simp2bi 1004	. . . . . . . . . . . . . 14 |- ( I e. ( 0..^ L ) -> L e. NN )
118	36, 117	syl 16	. . . . . . . . . . . . 13 |- ( ph -> L e. NN )
119	37, 118	eqeltrd 2539	. . . . . . . . . . . 12 |- ( ph -> ( # ` W ) e. NN )
120		wrdfin 12359	. . . . . . . . . . . . 13 |- ( W e. Word T -> W e. Fin )
121		hashnncl 12244	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
122	24, 120, 121	3syl 20	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
123	119, 122	mpbid 210	. . . . . . . . . . 11 |- ( ph -> W =/= (/) )
124	123	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W =/= (/) )
125		wrdeqcats1 12479	. . . . . . . . . 10 |- ( ( W e. Word T /\ W =/= (/) ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) )
126	115, 124, 125	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) concat <" ( W ` ( ( # ` W ) - 1 ) ) "> ) )
127	37	oveq1d 6208	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
128	127	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
129	118	nncnd 10442	. . . . . . . . . . . . 13 |- ( ph -> L e. CC )
130		ax-1cn 9444	. . . . . . . . . . . . . 14 |- 1 e. CC
131	130	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 1 e. CC )
132		elfzoelz 11663	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) -> I e. ZZ )
133	36, 132	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> I e. ZZ )
134	133	zcnd 10852	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
135	129, 131, 134	subadd2d 9842	. . . . . . . . . . . 12 |- ( ph -> ( ( L - 1 ) = I <-> ( I + 1 ) = L ) )
136	135	biimpar 485	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( L - 1 ) = I )
137	128, 136	eqtrd 2492	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = I )
138		opeq2 4161	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> <. 0 , ( ( # ` W ) - 1 ) >. = <. 0 , I >. )
139	138	oveq2d 6209	. . . . . . . . . . 11 |- ( ( ( # ` W ) - 1 ) = I -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
140		fveq2 5792	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` I ) )
141	140	s1eqd 12403	. . . . . . . . . . 11 |- ( ( ( # ` W ) - 1 ) = I -> <" ( W ` ( ( # ` W ) - 1 ) ) "> = <" ( W ` I ) "> )
142	139, 141	oveq12d 6211	. . . . . . . . . 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 2492	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) )
145	144	oveq2d 6209	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( G gsumg ( ( W substr <. 0 , I >. ) concat <" ( W ` I ) "> ) ) )
146	41	s1cld 12405	. . . . . . . . 9 |- ( ph -> <" ( W ` I ) "> e. Word ( Base ` G ) )
147		eqid 2451	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
148	20, 147	gsumccat 15630	. . . . . . . . 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 1219	. . . . . . . 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 465	. . . . . . 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 15628	. . . . . . . . . . 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 6209	. . . . . . . . 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 16006	. . . . . . . . . 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 661	. . . . . . . . 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 2492	. . . . . . . 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 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 ) ) )
158	145, 150, 157	3eqtrd 2496	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
159	158	difeq1d 3574	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg W ) \ _I ) = ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
160	159	dmeqd 5143	. . . 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 2542	. . 3 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( G gsumg W ) \ _I ) )
162	13, 161	mtand 659	. 2 |- ( ph -> -. ( I + 1 ) = L )
163		fzostep1 11745	. . . 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 377	. 2 |- ( ph -> ( -. ( I + 1 ) e. ( 0..^ L ) -> ( I + 1 ) = L ) )
166	162, 165	mt3d 125	1 |- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/_ wxo 1351   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   {crab 2799   _Vcvv 3071   \ cdif 3426   i^i cin 3428   C_ wss 3429   (/)c0 3738   {csn 3978   <.cop 3984   class class class wbr 4393   |-> cmpt 4451   _I cid 4732   dom cdm 4941   ran crn 4942   |` cres 4943   o. ccom 4945   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Fincfn 7413   CCcc 9384   0cc0 9386   1c1 9387   + caddc 9389   < clt 9522   - cmin 9699   NNcn 10426   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547  ..^cfzo 11658   #chash 12213  Word cword 12332   concat cconcat 12334   <"cs1 12335   substr csubstr 12336   Basecbs 14285   +g cplusg 14349   gsum_g cgsu 14490   Mndcmnd 15520   Grpcgrp 15521  SubMndcsubmnd 15574  SubGrpcsubg 15786   SymGrpcsymg 15993  pmTrspcpmtr 16058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-tset 14368  df-0g 14491  df-gsum 14492  df-mnd 15526  df-submnd 15576  df-grp 15656  df-minusg 15657  df-subg 15789  df-symg 15994  df-pmtr 16059

This theorem is referenced by:  psgnunilem2  16112