Theorem psgnfval 16011

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 5696	. . . . . . . . . 10 |- ( d = D -> ( SymGrp ` d ) = ( SymGrp ` D ) )
3		psgnfval.g	. . . . . . . . . 10 |- G = ( SymGrp ` D )
4	2, 3	syl6eqr 2493	. . . . . . . . 9 |- ( d = D -> ( SymGrp ` d ) = G )
5	4	fveq2d 5700	. . . . . . . 8 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = ( Base ` G ) )
6		psgnfval.b	. . . . . . . 8 |- B = ( Base ` G )
7	5, 6	syl6eqr 2493	. . . . . . 7 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = B )
8		rabeq 2971	. . . . . . 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 2493	. . . . 5 |- ( d = D -> { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } = F )
12		fveq2 5696	. . . . . . . . . 10 |- ( d = D -> (pmTrsp ` d ) = (pmTrsp ` D ) )
13	12	rneqd 5072	. . . . . . . . 9 |- ( d = D -> ran (pmTrsp ` d ) = ran (pmTrsp ` D ) )
14		psgnfval.t	. . . . . . . . 9 |- T = ran (pmTrsp ` D )
15	13, 14	syl6eqr 2493	. . . . . . . 8 |- ( d = D -> ran (pmTrsp ` d ) = T )
16		wrdeq 12256	. . . . . . . 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 6111	. . . . . . . . 9 |- ( d = D -> ( ( SymGrp ` d ) gsumg w ) = ( G gsumg w ) )
19	18	eqeq2d 2454	. . . . . . . 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 2937	. . . . . 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 5407	. . . . 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 4375	. . . 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 16002	. . . 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 5706	. . . . . . . 8 |- ( Base ` G ) e. _V
26	6, 25	eqeltri 2513	. . . . . . 7 |- B e. _V
27	26	rabex 4448	. . . . . 6 |- { p e. B | dom ( p \ _I ) e. Fin } e. _V
28	10, 27	eqeltri 2513	. . . . 5 |- F e. _V
29	28	mptex 5953	. . . 4 |- ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) e. _V
30	23, 24, 29	fvmpt 5779	. . 3 |- ( D e. _V -> (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
31		fvprc 5690	. . . 4 |- ( -. D e. _V -> (pmSgn ` D ) = (/) )
32		fvprc 5690	. . . . . . . . . . . . 13 |- ( -. D e. _V -> ( SymGrp ` D ) = (/) )
33	3, 32	syl5eq 2487	. . . . . . . . . . . 12 |- ( -. D e. _V -> G = (/) )
34	33	fveq2d 5700	. . . . . . . . . . 11 |- ( -. D e. _V -> ( Base ` G ) = ( Base ` (/) ) )
35		base0 14218	. . . . . . . . . . 11 |- (/) = ( Base ` (/) )
36	34, 35	syl6eqr 2493	. . . . . . . . . 10 |- ( -. D e. _V -> ( Base ` G ) = (/) )
37	6, 36	syl5eq 2487	. . . . . . . . 9 |- ( -. D e. _V -> B = (/) )
38		rabeq 2971	. . . . . . . . 9 |- ( B = (/) -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
39	37, 38	syl 16	. . . . . . . 8 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
40		rab0 3663	. . . . . . . 8 |- { p e. (/) | dom ( p \ _I ) e. Fin } = (/)
41	39, 40	syl6eq 2491	. . . . . . 7 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = (/) )
42	10, 41	syl5eq 2487	. . . . . 6 |- ( -. D e. _V -> F = (/) )
43	42	mpteq1d 4378	. . . . 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 ) ) ) ) ) )
44		mpt0 5543	. . . . 5 |- ( x e. (/) |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = (/)
45	43, 44	syl6eq 2491	. . . 4 |- ( -. D e. _V -> ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = (/) )
46	31, 45	eqtr4d 2478	. . 3 |- ( -. D e. _V -> (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) )
47	30, 46	pm2.61i 164	. 2 |- (pmSgn ` D ) = ( x e. F |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
48	1, 47	eqtri 2463	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 1369   e. wcel 1756   E.wrex 2721   {crab 2724   _Vcvv 2977   \ cdif 3330   (/)c0 3642   e. cmpt 4355   _I cid 4636   dom cdm 4845   ran crn 4846   iotacio 5384   ` cfv 5423  (class class class)co 6096   Fincfn 7315   1c1 9288   -ucneg 9601   ^cexp 11870   #chash 12108  Word cword 12226   Basecbs 14179   gsum_g cgsu 14384   SymGrpcsymg 15887  pmTrspcpmtr 15952  pmSgncpsgn 16000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-word 12234  df-slot 14183  df-base 14184  df-psgn 16002

This theorem is referenced by:  psgnfn  16012  psgnval  16018  psgnfvalfi  16024