Theorem psgndif 19948

Description: Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.)

Hypotheses

Ref	Expression
psgndif.p	|- P = ( Base ` ( SymGrp ` N ) )
psgndif.s	|- S = (pmSgn ` N )
psgndif.z	|- Z = (pmSgn ` ( N \ { K } ) )

Assertion

Ref	Expression
psgndif	|- ( ( N e. Fin /\ K e. N ) -> ( Q e. { q e. P | ( q ` K ) = K } -> ( Z ` ( Q |` ( N \ { K } ) ) ) = ( S ` Q ) ) )

Distinct variable groups:   K, q   P, q   Q, q

Allowed substitution hints:   S( q)   N( q)   Z( q)

Proof of Theorem psgndif

Dummy variables r s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgndif.p	. . . . . . . . . . 11 |- P = ( Base ` ( SymGrp ` N ) )
2		eqid 2622	. . . . . . . . . . 11 |- ran (pmTrsp ` ( N \ { K } ) ) = ran (pmTrsp ` ( N \ { K } ) )
3		eqid 2622	. . . . . . . . . . 11 |- ( SymGrp ` ( N \ { K } ) ) = ( SymGrp ` ( N \ { K } ) )
4		eqid 2622	. . . . . . . . . . 11 |- ( SymGrp ` N ) = ( SymGrp ` N )
5		eqid 2622	. . . . . . . . . . 11 |- ran (pmTrsp ` N ) = ran (pmTrsp ` N )
6	1, 2, 3, 4, 5	psgnfix2 19945	. . . . . . . . . 10 |- ( ( N e. Fin /\ K e. N ) -> ( Q e. { q e. P | ( q ` K ) = K } -> E. r e. Word ran (pmTrsp ` N ) Q = ( ( SymGrp ` N ) gsumg r ) ) )
7	6	imp 445	. . . . . . . . 9 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> E. r e. Word ran (pmTrsp ` N ) Q = ( ( SymGrp ` N ) gsumg r ) )
8	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) -> E. r e. Word ran (pmTrsp ` N ) Q = ( ( SymGrp ` N ) gsumg r ) )
9	1, 2, 3, 4, 5	psgndiflemA 19947	. . . . . . . . . . . . 13 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( ( w e. Word ran (pmTrsp ` ( N \ { K } ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( Q = ( ( SymGrp ` N ) gsumg r ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) ) )
10	9	imp 445	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ ( w e. Word ran (pmTrsp ` ( N \ { K } ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ r e. Word ran (pmTrsp ` N ) ) ) -> ( Q = ( ( SymGrp ` N ) gsumg r ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
11	10	3anassrs 1290	. . . . . . . . . . 11 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( Q = ( ( SymGrp ` N ) gsumg r ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
12	11	adantlrr 757	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( Q = ( ( SymGrp ` N ) gsumg r ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
13		eqeq1 2626	. . . . . . . . . . . 12 |- ( s = ( -u 1 ^ ( # ` w ) ) -> ( s = ( -u 1 ^ ( # ` r ) ) <-> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
14	13	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) -> ( s = ( -u 1 ^ ( # ` r ) ) <-> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
15	14	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( s = ( -u 1 ^ ( # ` r ) ) <-> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) ) )
16	12, 15	sylibrd 249	. . . . . . . . 9 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( Q = ( ( SymGrp ` N ) gsumg r ) -> s = ( -u 1 ^ ( # ` r ) ) ) )
17	16	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) -> A. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) -> s = ( -u 1 ^ ( # ` r ) ) ) )
18	8, 17	r19.29imd 3074	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) -> E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) )
19	18	ex 450	. . . . . 6 |- ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> ( ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) -> E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
20	19	rexlimdva 3031	. . . . 5 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) -> E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
21	1, 2, 3	psgnfix1 19944	. . . . . . . . . 10 |- ( ( N e. Fin /\ K e. N ) -> ( Q e. { q e. P | ( q ` K ) = K } -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) )
22	21	imp 445	. . . . . . . . 9 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) )
23	22	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) )
24		simp-4l 806	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) )
25		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> w e. Word ran (pmTrsp ` ( N \ { K } ) ) )
26	25	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> w e. Word ran (pmTrsp ` ( N \ { K } ) ) )
27		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) )
28		simp-4r 807	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> r e. Word ran (pmTrsp ` N ) )
29	26, 27, 28	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> ( w e. Word ran (pmTrsp ` ( N \ { K } ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ r e. Word ran (pmTrsp ` N ) ) )
30		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) -> Q = ( ( SymGrp ` N ) gsumg r ) )
31	30	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> Q = ( ( SymGrp ` N ) gsumg r ) )
32	24, 29, 31, 9	syl3c 66	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` r ) ) )
33	32	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) /\ ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) ) -> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) )
34	33	ex 450	. . . . . . . . . . 11 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ Q = ( ( SymGrp ` N ) gsumg r ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) -> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) ) )
35	34	adantlrr 757	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) -> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) ) )
36		eqeq1 2626	. . . . . . . . . . . 12 |- ( s = ( -u 1 ^ ( # ` r ) ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) ) )
37	36	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) ) )
38	37	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` r ) ) = ( -u 1 ^ ( # ` w ) ) ) )
39	35, 38	sylibrd 249	. . . . . . . . 9 |- ( ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) /\ w e. Word ran (pmTrsp ` ( N \ { K } ) ) ) -> ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) -> s = ( -u 1 ^ ( # ` w ) ) ) )
40	39	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) -> A. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) -> s = ( -u 1 ^ ( # ` w ) ) ) )
41	23, 40	r19.29imd 3074	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) /\ ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
42	41	ex 450	. . . . . 6 |- ( ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) /\ r e. Word ran (pmTrsp ` N ) ) -> ( ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
43	42	rexlimdva 3031	. . . . 5 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) -> E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
44	20, 43	impbid 202	. . . 4 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
45	44	iotabidv 5872	. . 3 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( iota s E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) = ( iota s E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
46		diffi 8192	. . . . 5 |- ( N e. Fin -> ( N \ { K } ) e. Fin )
47	46	ad2antrr 762	. . . 4 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( N \ { K } ) e. Fin )
48		eqid 2622	. . . . . 6 |- { q e. P | ( q ` K ) = K } = { q e. P | ( q ` K ) = K }
49		eqid 2622	. . . . . 6 |- ( Base ` ( SymGrp ` ( N \ { K } ) ) ) = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
50		eqid 2622	. . . . . 6 |- ( N \ { K } ) = ( N \ { K } )
51	1, 48, 49, 50	symgfixelsi 17855	. . . . 5 |- ( ( K e. N /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( Q |` ( N \ { K } ) ) e. ( Base ` ( SymGrp ` ( N \ { K } ) ) ) )
52	51	adantll 750	. . . 4 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( Q |` ( N \ { K } ) ) e. ( Base ` ( SymGrp ` ( N \ { K } ) ) ) )
53		psgndif.z	. . . . 5 |- Z = (pmSgn ` ( N \ { K } ) )
54	3, 49, 2, 53	psgnvalfi 17934	. . . 4 |- ( ( ( N \ { K } ) e. Fin /\ ( Q |` ( N \ { K } ) ) e. ( Base ` ( SymGrp ` ( N \ { K } ) ) ) ) -> ( Z ` ( Q |` ( N \ { K } ) ) ) = ( iota s E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
55	47, 52, 54	syl2anc 693	. . 3 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( Z ` ( Q |` ( N \ { K } ) ) ) = ( iota s E. w e. Word ran (pmTrsp ` ( N \ { K } ) ) ( ( Q |` ( N \ { K } ) ) = ( ( SymGrp ` ( N \ { K } ) ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
56		simpl 473	. . . 4 |- ( ( N e. Fin /\ K e. N ) -> N e. Fin )
57		elrabi 3359	. . . 4 |- ( Q e. { q e. P | ( q ` K ) = K } -> Q e. P )
58		psgndif.s	. . . . 5 |- S = (pmSgn ` N )
59	4, 1, 5, 58	psgnvalfi 17934	. . . 4 |- ( ( N e. Fin /\ Q e. P ) -> ( S ` Q ) = ( iota s E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
60	56, 57, 59	syl2an 494	. . 3 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( S ` Q ) = ( iota s E. r e. Word ran (pmTrsp ` N ) ( Q = ( ( SymGrp ` N ) gsumg r ) /\ s = ( -u 1 ^ ( # ` r ) ) ) ) )
61	45, 55, 60	3eqtr4d 2666	. 2 |- ( ( ( N e. Fin /\ K e. N ) /\ Q e. { q e. P | ( q ` K ) = K } ) -> ( Z ` ( Q |` ( N \ { K } ) ) ) = ( S ` Q ) )
62	61	ex 450	1 |- ( ( N e. Fin /\ K e. N ) -> ( Q e. { q e. P | ( q ` K ) = K } -> ( Z ` ( Q |` ( N \ { K } ) ) ) = ( S ` Q ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   \ cdif 3571   {csn 4177   ran crn 5115   |` cres 5116   iotacio 5849   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   -ucneg 10267   ^cexp 12860   #chash 13117  Word cword 13291   Basecbs 15857   gsum_g cgsu 16101   SymGrpcsymg 17797  pmTrspcpmtr 17861  pmSgncpsgn 17909

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-iin 4523  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-tpos 7352  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-xnn0 11364  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-reverse 13305  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-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-gim 17701  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911

This theorem is referenced by:  zrhcopsgndif  19949