Theorem psgnunilem4 17917

Description: Lemma for psgnuni 17919. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Hypotheses

Ref	Expression
psgnunilem4.g	|- G = ( SymGrp ` D )
psgnunilem4.t	|- T = ran (pmTrsp ` D )
psgnunilem4.d	|- ( ph -> D e. V )
psgnunilem4.w1	|- ( ph -> W e. Word T )
psgnunilem4.w2	|- ( ph -> ( G gsumg W ) = ( _I |` D ) )

Assertion

Ref	Expression
psgnunilem4	|- ( ph -> ( -u 1 ^ ( # ` W ) ) = 1 )

Proof of Theorem psgnunilem4

Dummy variables x w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnunilem4.w1	. 2 |- ( ph -> W e. Word T )
2		psgnunilem4.w2	. 2 |- ( ph -> ( G gsumg W ) = ( _I |` D ) )
3		wrdfin 13323	. . . . 5 |- ( W e. Word T -> W e. Fin )
4		hashcl 13147	. . . . 5 |- ( W e. Fin -> ( # ` W ) e. NN0 )
5	1, 3, 4	3syl 18	. . . 4 |- ( ph -> ( # ` W ) e. NN0 )
6		nn0uz 11722	. . . 4 |- NN0 = ( ZZ>= ` 0 )
7	5, 6	syl6eleq 2711	. . 3 |- ( ph -> ( # ` W ) e. ( ZZ>= ` 0 ) )
8		fveq2 6191	. . . . . . . . 9 |- ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
9		hash0 13158	. . . . . . . . 9 |- ( # ` (/) ) = 0
10	8, 9	syl6eq 2672	. . . . . . . 8 |- ( w = (/) -> ( # ` w ) = 0 )
11	10	oveq2d 6666	. . . . . . 7 |- ( w = (/) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ 0 ) )
12		neg1cn 11124	. . . . . . . 8 |- -u 1 e. CC
13		exp0 12864	. . . . . . . 8 |- ( -u 1 e. CC -> ( -u 1 ^ 0 ) = 1 )
14	12, 13	ax-mp 5	. . . . . . 7 |- ( -u 1 ^ 0 ) = 1
15	11, 14	syl6eq 2672	. . . . . 6 |- ( w = (/) -> ( -u 1 ^ ( # ` w ) ) = 1 )
16	15	2a1d 26	. . . . 5 |- ( w = (/) -> ( ( ph /\ A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
17		psgnunilem4.g	. . . . . . . . . . . . 13 |- G = ( SymGrp ` D )
18		psgnunilem4.t	. . . . . . . . . . . . 13 |- T = ran (pmTrsp ` D )
19		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ph )
20		psgnunilem4.d	. . . . . . . . . . . . . 14 |- ( ph -> D e. V )
21	19, 20	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> D e. V )
22		simpl3l 1116	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> w e. Word T )
23		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( # ` w ) = ( # ` w ) )
24		wrdfin 13323	. . . . . . . . . . . . . . 15 |- ( w e. Word T -> w e. Fin )
25	22, 24	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> w e. Fin )
26		simpl2 1065	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> w =/= (/) )
27		hashnncl 13157	. . . . . . . . . . . . . . 15 |- ( w e. Fin -> ( ( # ` w ) e. NN <-> w =/= (/) ) )
28	27	biimpar 502	. . . . . . . . . . . . . 14 |- ( ( w e. Fin /\ w =/= (/) ) -> ( # ` w ) e. NN )
29	25, 26, 28	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( # ` w ) e. NN )
30		simpl3r 1117	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( G gsumg w ) = ( _I |` D ) )
31		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> ( # ` x ) = ( # ` y ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( ( # ` x ) = ( ( # ` w ) - 2 ) <-> ( # ` y ) = ( ( # ` w ) - 2 ) ) )
33		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> ( G gsumg x ) = ( G gsumg y ) )
34	33	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( ( G gsumg x ) = ( _I |` D ) <-> ( G gsumg y ) = ( _I |` D ) ) )
35	32, 34	anbi12d 747	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) <-> ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsumg y ) = ( _I |` D ) ) ) )
36	35	cbvrexv 3172	. . . . . . . . . . . . . . . 16 |- ( E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) <-> E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsumg y ) = ( _I |` D ) ) )
37	36	notbii 310	. . . . . . . . . . . . . . 15 |- ( -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) <-> -. E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsumg y ) = ( _I |` D ) ) )
38	37	biimpi 206	. . . . . . . . . . . . . 14 |- ( -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) -> -. E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsumg y ) = ( _I |` D ) ) )
39	38	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> -. E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsumg y ) = ( _I |` D ) ) )
40	17, 18, 21, 22, 23, 29, 30, 39	psgnunilem3 17916	. . . . . . . . . . . 12 |- -. ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) )
41		iman 440	. . . . . . . . . . . 12 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) <-> -. ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) )
42	40, 41	mpbir 221	. . . . . . . . . . 11 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) )
43		df-rex 2918	. . . . . . . . . . 11 |- ( E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) <-> E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) )
44	42, 43	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) )
45		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> x e. Word T )
46		simprrr 805	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( G gsumg x ) = ( _I |` D ) )
47	45, 46	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) )
48		wrdfin 13323	. . . . . . . . . . . . . . . . . 18 |- ( x e. Word T -> x e. Fin )
49		hashcl 13147	. . . . . . . . . . . . . . . . . 18 |- ( x e. Fin -> ( # ` x ) e. NN0 )
50	45, 48, 49	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` x ) e. NN0 )
51		simp3l 1089	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> w e. Word T )
52	51, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> w e. Fin )
53		simp2 1062	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> w =/= (/) )
54	52, 53, 28	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( # ` w ) e. NN )
55	54	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. NN )
56		simprrl 804	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` x ) = ( ( # ` w ) - 2 ) )
57	55	nnred 11035	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. RR )
58		2rp 11837	. . . . . . . . . . . . . . . . . . 19 |- 2 e. RR+
59		ltsubrp 11866	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` w ) e. RR /\ 2 e. RR+ ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
60	57, 58, 59	sylancl 694	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
61	56, 60	eqbrtrd 4675	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` x ) < ( # ` w ) )
62		elfzo0 12508	. . . . . . . . . . . . . . . . 17 |- ( ( # ` x ) e. ( 0..^ ( # ` w ) ) <-> ( ( # ` x ) e. NN0 /\ ( # ` w ) e. NN /\ ( # ` x ) < ( # ` w ) ) )
63	50, 55, 61, 62	syl3anbrc 1246	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` x ) e. ( 0..^ ( # ` w ) ) )
64		id 22	. . . . . . . . . . . . . . . . 17 |- ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
65	64	com13 88	. . . . . . . . . . . . . . . 16 |- ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
66	47, 63, 65	sylc 65	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) )
67	56	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` x ) ) = ( -u 1 ^ ( ( # ` w ) - 2 ) ) )
68	12	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> -u 1 e. CC )
69		neg1ne0 11126	. . . . . . . . . . . . . . . . . . 19 |- -u 1 =/= 0
70	69	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> -u 1 =/= 0 )
71		2z 11409	. . . . . . . . . . . . . . . . . . 19 |- 2 e. ZZ
72	71	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> 2 e. ZZ )
73	55	nnzd 11481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. ZZ )
74	68, 70, 72, 73	expsubd 13019	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( ( # ` w ) - 2 ) ) = ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) )
75		neg1sqe1 12959	. . . . . . . . . . . . . . . . . . 19 |- ( -u 1 ^ 2 ) = 1
76	75	oveq2i 6661	. . . . . . . . . . . . . . . . . 18 |- ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) = ( ( -u 1 ^ ( # ` w ) ) / 1 )
77		m1expcl 12883	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` w ) e. ZZ -> ( -u 1 ^ ( # ` w ) ) e. ZZ )
78	77	zcnd 11483	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` w ) e. ZZ -> ( -u 1 ^ ( # ` w ) ) e. CC )
79	73, 78	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` w ) ) e. CC )
80	79	div1d 10793	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` w ) ) / 1 ) = ( -u 1 ^ ( # ` w ) ) )
81	76, 80	syl5eq 2668	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) = ( -u 1 ^ ( # ` w ) ) )
82	67, 74, 81	3eqtrd 2660	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` x ) ) = ( -u 1 ^ ( # ` w ) ) )
83	82	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` x ) ) = 1 <-> ( -u 1 ^ ( # ` w ) ) = 1 ) )
84	66, 83	sylibd 229	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) ) -> ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
85	84	ex 450	. . . . . . . . . . . . 13 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
86	85	com23 86	. . . . . . . . . . . 12 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
87	86	alimdv 1845	. . . . . . . . . . 11 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> A. x ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
88		19.23v 1902	. . . . . . . . . . 11 |- ( A. x ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
89	87, 88	syl6ib 241	. . . . . . . . . 10 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsumg x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
90	44, 89	mpid 44	. . . . . . . . 9 |- ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
91	90	3exp 1264	. . . . . . . 8 |- ( ph -> ( w =/= (/) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
92	91	com34 91	. . . . . . 7 |- ( ph -> ( w =/= (/) -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
93	92	com12 32	. . . . . 6 |- ( w =/= (/) -> ( ph -> ( A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
94	93	impd 447	. . . . 5 |- ( w =/= (/) -> ( ( ph /\ A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
95	16, 94	pm2.61ine 2877	. . . 4 |- ( ( ph /\ A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
96	95	3adant2 1080	. . 3 |- ( ( ph /\ ( # ` w ) e. ( 0 ... ( # ` W ) ) /\ A. x ( ( # ` x ) e. ( 0..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
97		eleq1 2689	. . . . 5 |- ( w = x -> ( w e. Word T <-> x e. Word T ) )
98		oveq2 6658	. . . . . 6 |- ( w = x -> ( G gsumg w ) = ( G gsumg x ) )
99	98	eqeq1d 2624	. . . . 5 |- ( w = x -> ( ( G gsumg w ) = ( _I |` D ) <-> ( G gsumg x ) = ( _I |` D ) ) )
100	97, 99	anbi12d 747	. . . 4 |- ( w = x -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) <-> ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) ) )
101		fveq2 6191	. . . . . 6 |- ( w = x -> ( # ` w ) = ( # ` x ) )
102	101	oveq2d 6666	. . . . 5 |- ( w = x -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` x ) ) )
103	102	eqeq1d 2624	. . . 4 |- ( w = x -> ( ( -u 1 ^ ( # ` w ) ) = 1 <-> ( -u 1 ^ ( # ` x ) ) = 1 ) )
104	100, 103	imbi12d 334	. . 3 |- ( w = x -> ( ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( ( x e. Word T /\ ( G gsumg x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
105		eleq1 2689	. . . . 5 |- ( w = W -> ( w e. Word T <-> W e. Word T ) )
106		oveq2 6658	. . . . . 6 |- ( w = W -> ( G gsumg w ) = ( G gsumg W ) )
107	106	eqeq1d 2624	. . . . 5 |- ( w = W -> ( ( G gsumg w ) = ( _I |` D ) <-> ( G gsumg W ) = ( _I |` D ) ) )
108	105, 107	anbi12d 747	. . . 4 |- ( w = W -> ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) <-> ( W e. Word T /\ ( G gsumg W ) = ( _I |` D ) ) ) )
109		fveq2 6191	. . . . . 6 |- ( w = W -> ( # ` w ) = ( # ` W ) )
110	109	oveq2d 6666	. . . . 5 |- ( w = W -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` W ) ) )
111	110	eqeq1d 2624	. . . 4 |- ( w = W -> ( ( -u 1 ^ ( # ` w ) ) = 1 <-> ( -u 1 ^ ( # ` W ) ) = 1 ) )
112	108, 111	imbi12d 334	. . 3 |- ( w = W -> ( ( ( w e. Word T /\ ( G gsumg w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( ( W e. Word T /\ ( G gsumg W ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` W ) ) = 1 ) ) )
113	1, 7, 96, 104, 112, 101, 109	uzindi 12781	. 2 |- ( ph -> ( ( W e. Word T /\ ( G gsumg W ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` W ) ) = 1 ) )
114	1, 2, 113	mp2and 715	1 |- ( ph -> ( -u 1 ^ ( # ` W ) ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915   class class class wbr 4653   _I cid 5023   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   ^cexp 12860   #chash 13117  Word cword 13291   gsum_g cgsu 16101   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-ot 4186  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-se 5074  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-isom 5897  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-div 10685  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-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  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:  psgnuni  17919