Theorem psgnfval 16652

Description: Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)

Hypotheses

Ref	Expression
psgnfval.g	|- G = ( SymGrp ` D )
psgnfval.b	|- B = ( Base ` G )
psgnfval.f	|- F = { p e. B | dom ( p \ _I ) e. Fin }
psgnfval.t	|- T = ran (pmTrsp ` D )
psgnfval.n	|- N = (pmSgn ` D )

Assertion

Ref	Expression
psgnfval	|- N = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )

Distinct variable groups:   s, p, w, x   D, s, w, x   x, F   w, T   B, p

Allowed substitution hints:   B( x, w, s)   D( p)   T( x, s, p)   F( w, s, p)   G( x, w, s, p)   N( x, w, s, p)

Proof of Theorem psgnfval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnfval.n	. 2 |- N = (pmSgn ` D )
2		fveq2 5872	. . . . . . . . . 10 |- ( d = D -> ( SymGrp ` d ) = ( SymGrp ` D ) )
3		psgnfval.g	. . . . . . . . . 10 |- G = ( SymGrp ` D )
4	2, 3	syl6eqr 2516	. . . . . . . . 9 |- ( d = D -> ( SymGrp ` d ) = G )
5	4	fveq2d 5876	. . . . . . . 8 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = ( Base ` G ) )
6		psgnfval.b	. . . . . . . 8 |- B = ( Base ` G )
7	5, 6	syl6eqr 2516	. . . . . . 7 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = B )
8		rabeq 3103	. . . . . . 7 |- ( ( Base ` ( SymGrp ` d ) ) = B -> { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } = { p e. B | dom ( p \ _I ) e. Fin } )
9	7, 8	syl 16	. . . . . 6 |- ( d = D -> { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } = { p e. B | dom ( p \ _I ) e. Fin } )
10		psgnfval.f	. . . . . 6 |- F = { p e. B | dom ( p \ _I ) e. Fin }
11	9, 10	syl6eqr 2516	. . . . 5 |- ( d = D -> { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } = F )
12		fveq2 5872	. . . . . . . . . 10 |- ( d = D -> (pmTrsp ` d ) = (pmTrsp ` D ) )
13	12	rneqd 5240	. . . . . . . . 9 |- ( d = D -> ran (pmTrsp ` d ) = ran (pmTrsp ` D ) )
14		psgnfval.t	. . . . . . . . 9 |- T = ran (pmTrsp ` D )
15	13, 14	syl6eqr 2516	. . . . . . . 8 |- ( d = D -> ran (pmTrsp ` d ) = T )
16		wrdeq 12571	. . . . . . . 8 |- ( ran (pmTrsp ` d ) = T -> Word ran (pmTrsp ` d ) = Word T )
17	15, 16	syl 16	. . . . . . 7 |- ( d = D -> Word ran (pmTrsp ` d ) = Word T )
18	4	oveq1d 6311	. . . . . . . . 9 |- ( d = D -> ( ( SymGrp ` d ) gsumg w ) = ( G gsumg w ) )
19	18	eqeq2d 2471	. . . . . . . 8 |- ( d = D -> ( x = ( ( SymGrp ` d ) gsumg w ) <-> x = ( G gsumg w ) ) )
20	19	anbi1d 704	. . . . . . 7 |- ( d = D -> ( ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
21	17, 20	rexeqbidv 3069	. . . . . 6 |- ( d = D -> ( E. w e. Word ran (pmTrsp ` d ) ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
22	21	iotabidv 5578	. . . . 5 |- ( d = D -> ( iota s E. w e. Word ran (pmTrsp ` d ) ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) = ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
23	11, 22	mpteq12dv 4535	. . . 4 |- ( d = D -> ( x e. { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } |-> ( iota s E. w e. Word ran (pmTrsp ` d ) ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
24		df-psgn 16643	. . . 4 |- pmSgn = ( d e. _V |-> ( x e. { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } |-> ( iota s E. w e. Word ran (pmTrsp ` d ) ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
25		fvex 5882	. . . . . . 7 |- ( Base ` G ) e. _V
26	6, 25	eqeltri 2541	. . . . . 6 |- B e. _V
27	10, 26	rabex2 4609	. . . . 5 |- F e. _V
28	27	mptex 6144	. . . 4 |- ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) e. _V
29	23, 24, 28	fvmpt 5956	. . 3 |- ( D e. _V -> (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
30		fvprc 5866	. . . 4 |- ( -. D e. _V -> (pmSgn ` D ) = (/) )
31		fvprc 5866	. . . . . . . . . . . . 13 |- ( -. D e. _V -> ( SymGrp ` D ) = (/) )
32	3, 31	syl5eq 2510	. . . . . . . . . . . 12 |- ( -. D e. _V -> G = (/) )
33	32	fveq2d 5876	. . . . . . . . . . 11 |- ( -. D e. _V -> ( Base ` G ) = ( Base ` (/) ) )
34		base0 14685	. . . . . . . . . . 11 |- (/) = ( Base ` (/) )
35	33, 34	syl6eqr 2516	. . . . . . . . . 10 |- ( -. D e. _V -> ( Base ` G ) = (/) )
36	6, 35	syl5eq 2510	. . . . . . . . 9 |- ( -. D e. _V -> B = (/) )
37		rabeq 3103	. . . . . . . . 9 |- ( B = (/) -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
38	36, 37	syl 16	. . . . . . . 8 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
39		rab0 3815	. . . . . . . 8 |- { p e. (/) | dom ( p \ _I ) e. Fin } = (/)
40	38, 39	syl6eq 2514	. . . . . . 7 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = (/) )
41	10, 40	syl5eq 2510	. . . . . 6 |- ( -. D e. _V -> F = (/) )
42	41	mpteq1d 4538	. . . . 5 |- ( -. D e. _V -> ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = ( x e. (/) |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
43		mpt0 5714	. . . . 5 |- ( x e. (/) |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = (/)
44	42, 43	syl6eq 2514	. . . 4 |- ( -. D e. _V -> ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = (/) )
45	30, 44	eqtr4d 2501	. . 3 |- ( -. D e. _V -> (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
46	29, 45	pm2.61i 164	. 2 |- (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
47	1, 46	eqtri 2486	1 |- N = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1395   e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109   \ cdif 3468   (/)c0 3793   |-> cmpt 4515   _I cid 4799   dom cdm 5008   ran crn 5009   iotacio 5555   ` cfv 5594  (class class class)co 6296   Fincfn 7535   1c1 9510   -ucneg 9825   ^cexp 12169   #chash 12408  Word cword 12538   Basecbs 14644   gsum_g cgsu 14858   SymGrpcsymg 16529  pmTrspcpmtr 16593  pmSgncpsgn 16641

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

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-slot 14648  df-base 14649  df-psgn 16643

This theorem is referenced by:  psgnfn  16653  psgnval  16659  psgnfvalfi  16665