Theorem psgnunilem5 17914

Description: Lemma for psgnuni 17919. 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 3919	. . . 4 |- -. A e. (/)
2		psgnunilem2.id	. . . . . . . 8 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3	2	difeq1d 3727	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) \ _I ) = ( ( _I |` D ) \ _I ) )
4	3	dmeqd 5326	. . . . . 6 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = dom ( ( _I |` D ) \ _I ) )
5		resss 5422	. . . . . . . . 9 |- ( _I |` D ) C_ _I
6		ssdif0 3942	. . . . . . . . 9 |- ( ( _I |` D ) C_ _I <-> ( ( _I |` D ) \ _I ) = (/) )
7	5, 6	mpbi 220	. . . . . . . 8 |- ( ( _I |` D ) \ _I ) = (/)
8	7	dmeqi 5325	. . . . . . 7 |- dom ( ( _I |` D ) \ _I ) = dom (/)
9		dm0 5339	. . . . . . 7 |- dom (/) = (/)
10	8, 9	eqtri 2644	. . . . . 6 |- dom ( ( _I |` D ) \ _I ) = (/)
11	4, 10	syl6eq 2672	. . . . 5 |- ( ph -> dom ( ( G gsumg W ) \ _I ) = (/) )
12	11	eleq2d 2687	. . . 4 |- ( ph -> ( A e. dom ( ( G gsumg W ) \ _I ) <-> A e. (/) ) )
13	1, 12	mtbiri 317	. . 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 17820	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
17		grpmnd 17429	. . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
21	19, 15, 20	symgtrf 17889	. . . . . . . . . . 11 |- T C_ ( Base ` G )
22		sswrd 13313	. . . . . . . . . . 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 3604	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
26		swrdcl 13419	. . . . . . . . 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 17377	. . . . . . . 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 693	. . . . . . 7 |- ( ph -> ( G gsumg ( W substr <. 0 , I >. ) ) e. ( Base ` G ) )
30	15, 20	symgbasf1o 17803	. . . . . . 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 481	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) : D -1-1-onto-> D )
33		wrdf 13310	. . . . . . . . . 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 6666	. . . . . . . . . 10 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ L ) )
38	35, 37	eleqtrrd 2704	. . . . . . . . 9 |- ( ph -> I e. ( 0..^ ( # ` W ) ) )
39	34, 38	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( W ` I ) e. T )
40	21, 39	sseldi 3601	. . . . . . 7 |- ( ph -> ( W ` I ) e. ( Base ` G ) )
41	15, 20	symgbasf1o 17803	. . . . . . 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 481	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W ` I ) : D -1-1-onto-> D )
44	15, 20	symgsssg 17887	. . . . . . . . . . . 12 |- ( D e. V -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubGrp ` G ) )
45		subgsubm 17616	. . . . . . . . . . . 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 481	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } e. (SubMnd ` G ) )
48		fzossfz 12488	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ L ) C_ ( 0 ... L )
49	48, 35	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> I e. ( 0 ... L ) )
50		elfzuz3 12339	. . . . . . . . . . . . . . . . . . . 20 |- ( I e. ( 0 ... L ) -> L e. ( ZZ>= ` I ) )
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ( ZZ>= ` I ) )
52	36, 51	eqeltrd 2701	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` W ) e. ( ZZ>= ` I ) )
53		fzoss2 12496	. . . . . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ I ) ) -> s e. ( 0..^ ( # ` W ) ) )
56	34	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. T )
57	21, 56	sseldi 3601	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0..^ ( # ` W ) ) ) -> ( W ` s ) e. ( Base ` G ) )
58	55, 57	syldan 487	. . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = s -> ( W ` k ) = ( W ` s ) )
61	60	difeq1d 3727	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = s -> ( ( W ` k ) \ _I ) = ( ( W ` s ) \ _I ) )
62	61	dmeqd 5326	. . . . . . . . . . . . . . . . . . . 20 |- ( k = s -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` s ) \ _I ) )
63	62	eleq2d 2687	. . . . . . . . . . . . . . . . . . 19 |- ( k = s -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` s ) \ _I ) ) )
64	63	notbid 308	. . . . . . . . . . . . . . . . . 18 |- ( k = s -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
65	64	cbvralv 3171	. . . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . . 16 |- ( ph -> A. s e. ( 0..^ I ) -. A e. dom ( ( W ` s ) \ _I ) )
67	66	r19.21bi 2932	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0..^ I ) ) -> -. A e. dom ( ( W ` s ) \ _I ) )
68		difeq1 3721	. . . . . . . . . . . . . . . . . . 19 |- ( j = ( W ` s ) -> ( j \ _I ) = ( ( W ` s ) \ _I ) )
69	68	dmeqd 5326	. . . . . . . . . . . . . . . . . 18 |- ( j = ( W ` s ) -> dom ( j \ _I ) = dom ( ( W ` s ) \ _I ) )
70	69	sseq1d 3632	. . . . . . . . . . . . . . . . 17 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) ) )
71		disj2 4024	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) )
72		disjsn 4246	. . . . . . . . . . . . . . . . . 18 |- ( ( dom ( ( W ` s ) \ _I ) i^i { A } ) = (/) <-> -. A e. dom ( ( W ` s ) \ _I ) )
73	71, 72	bitr3i 266	. . . . . . . . . . . . . . . . 17 |- ( dom ( ( W ` s ) \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) )
74	70, 73	syl6bb 276	. . . . . . . . . . . . . . . 16 |- ( j = ( W ` s ) -> ( dom ( j \ _I ) C_ ( _V \ { A } ) <-> -. A e. dom ( ( W ` s ) \ _I ) ) )
75	74	elrab 3363	. . . . . . . . . . . . . . 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 698	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0..^ I ) ) -> ( W ` s ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
77		eqid 2622	. . . . . . . . . . . . . 14 |- ( s e. ( 0..^ I ) |-> ( W ` s ) ) = ( s e. ( 0..^ I ) |-> ( W ` s ) )
78	76, 77	fmptd 6385	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0..^ I ) |-> ( W ` s ) ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
79	36	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) )
80	49, 79	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 |- ( ph -> I e. ( 0 ... ( # ` W ) ) )
81		swrd0val 13421	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word T /\ I e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
82	24, 80, 81	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( W substr <. 0 , I >. ) = ( W |` ( 0..^ I ) ) )
83	34	feqmptd 6249	. . . . . . . . . . . . . . . 16 |- ( ph -> W = ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) )
84	83	reseq1d 5395	. . . . . . . . . . . . . . 15 |- ( ph -> ( W |` ( 0..^ I ) ) = ( ( s e. ( 0..^ ( # ` W ) ) |-> ( W ` s ) ) |` ( 0..^ I ) ) )
85		resmpt 5449	. . . . . . . . . . . . . . . 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 2660	. . . . . . . . . . . . . 14 |- ( ph -> ( W substr <. 0 , I >. ) = ( s e. ( 0..^ I ) |-> ( W ` s ) ) )
88	87	feq1d 6030	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 12 |- ( ph -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
90	89	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( W substr <. 0 , I >. ) : ( 0..^ I ) --> { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
91		iswrdi 13309	. . . . . . . . . . 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 17375	. . . . . . . . . 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 693	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg ( W substr <. 0 , I >. ) ) e. { j e. ( Base ` G ) | dom ( j \ _I ) C_ ( _V \ { A } ) } )
95		difeq1 3721	. . . . . . . . . . . . . 14 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> ( j \ _I ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
96	95	dmeqd 5326	. . . . . . . . . . . . 13 |- ( j = ( G gsumg ( W substr <. 0 , I >. ) ) -> dom ( j \ _I ) = dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) )
97	96	sseq1d 3632	. . . . . . . . . . . 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 3363	. . . . . . . . . . 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 480	. . . . . . . . . 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 4024	. . . . . . . . . . 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 4246	. . . . . . . . . . 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 266	. . . . . . . . . 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 208	. . . . . . . . 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 481	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( W ` I ) \ _I ) )
107	104, 106	jca 554	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( -. A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) /\ A e. dom ( ( W ` I ) \ _I ) ) )
108	107	olcd 408	. . . . . 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 1469	. . . . . 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 224	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( A e. dom ( ( G gsumg ( W substr <. 0 , I >. ) ) \ _I ) \/_ A e. dom ( ( W ` I ) \ _I ) ) )
111		f1omvdco3 17869	. . . . 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 1326	. . . 4 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
113	24	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W e. Word T )
114		elfzo0 12508	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) <-> ( I e. NN0 /\ L e. NN /\ I < L ) )
115	114	simp2bi 1077	. . . . . . . . . . . . . 14 |- ( I e. ( 0..^ L ) -> L e. NN )
116	35, 115	syl 17	. . . . . . . . . . . . 13 |- ( ph -> L e. NN )
117	36, 116	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ph -> ( # ` W ) e. NN )
118		wrdfin 13323	. . . . . . . . . . . . 13 |- ( W e. Word T -> W e. Fin )
119		hashnncl 13157	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
120	24, 118, 119	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
121	117, 120	mpbid 222	. . . . . . . . . . 11 |- ( ph -> W =/= (/) )
122	121	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> W =/= (/) )
123		swrdccatwrd 13468	. . . . . . . . . . 11 |- ( ( W e. Word T /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = W )
124	123	eqcomd 2628	. . . . . . . . . 10 |- ( ( W e. Word T /\ W =/= (/) ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
125	113, 122, 124	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) )
126	36	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
127	126	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = ( L - 1 ) )
128	116	nncnd 11036	. . . . . . . . . . . . 13 |- ( ph -> L e. CC )
129		1cnd 10056	. . . . . . . . . . . . 13 |- ( ph -> 1 e. CC )
130		elfzoelz 12470	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ L ) -> I e. ZZ )
131	35, 130	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> I e. ZZ )
132	131	zcnd 11483	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
133	128, 129, 132	subadd2d 10411	. . . . . . . . . . . 12 |- ( ph -> ( ( L - 1 ) = I <-> ( I + 1 ) = L ) )
134	133	biimpar 502	. . . . . . . . . . 11 |- ( ( ph /\ ( I + 1 ) = L ) -> ( L - 1 ) = I )
135	127, 134	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( # ` W ) - 1 ) = I )
136		opeq2 4403	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> <. 0 , ( ( # ` W ) - 1 ) >. = <. 0 , I >. )
137	136	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( # ` W ) - 1 ) = I -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
138	137	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) = ( W substr <. 0 , I >. ) )
139		lsw 13351	. . . . . . . . . . . . . 14 |- ( W e. Word T -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
140	24, 139	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
141		fveq2 6191	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) - 1 ) = I -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` I ) )
142	140, 141	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( lastS ` W ) = ( W ` I ) )
143	142	s1eqd 13381	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> <" ( lastS ` W ) "> = <" ( W ` I ) "> )
144	138, 143	oveq12d 6668	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` W ) - 1 ) = I ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
145	135, 144	syldan 487	. . . . . . . . 9 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
146	125, 145	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( I + 1 ) = L ) -> W = ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) )
147	146	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( G gsumg ( ( W substr <. 0 , I >. ) ++ <" ( W ` I ) "> ) ) )
148	40	s1cld 13383	. . . . . . . . 9 |- ( ph -> <" ( W ` I ) "> e. Word ( Base ` G ) )
149		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
150	20, 149	gsumccat 17378	. . . . . . . . 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 1326	. . . . . . . 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 481	. . . . . . 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 17376	. . . . . . . . . . 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 6666	. . . . . . . . 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 17810	. . . . . . . . . 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 693	. . . . . . . . 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 2656	. . . . . . . 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 481	. . . . . . 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 2660	. . . . . 6 |- ( ( ph /\ ( I + 1 ) = L ) -> ( G gsumg W ) = ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) )
161	160	difeq1d 3727	. . . . 5 |- ( ( ph /\ ( I + 1 ) = L ) -> ( ( G gsumg W ) \ _I ) = ( ( ( G gsumg ( W substr <. 0 , I >. ) ) o. ( W ` I ) ) \ _I ) )
162	161	dmeqd 5326	. . . 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 2704	. . 3 |- ( ( ph /\ ( I + 1 ) = L ) -> A e. dom ( ( G gsumg W ) \ _I ) )
164	13, 163	mtand 691	. 2 |- ( ph -> -. ( I + 1 ) = L )
165		fzostep1 12584	. . . 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 392	. 2 |- ( ph -> ( -. ( I + 1 ) e. ( 0..^ L ) -> ( I + 1 ) = L ) )
168	164, 167	mt3d 140	1 |- ( ph -> ( I + 1 ) e. ( 0..^ L ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/_ wxo 1464   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334   Grpcgrp 17422  SubGrpcsubg 17588   SymGrpcsymg 17797  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-symg 17798  df-pmtr 17862

This theorem is referenced by:  psgnunilem2  17915