Theorem psgnfval 17153

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 5870	. . . . . . . . . 10 |- ( d = D -> ( SymGrp ` d ) = ( SymGrp ` D ) )
3		psgnfval.g	. . . . . . . . . 10 |- G = ( SymGrp ` D )
4	2, 3	syl6eqr 2505	. . . . . . . . 9 |- ( d = D -> ( SymGrp ` d ) = G )
5	4	fveq2d 5874	. . . . . . . 8 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = ( Base ` G ) )
6		psgnfval.b	. . . . . . . 8 |- B = ( Base ` G )
7	5, 6	syl6eqr 2505	. . . . . . 7 |- ( d = D -> ( Base ` ( SymGrp ` d ) ) = B )
8		rabeq 3040	. . . . . . 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 17	. . . . . 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 2505	. . . . 5 |- ( d = D -> { p e. ( Base ` ( SymGrp ` d ) ) | dom ( p \ _I ) e. Fin } = F )
12		fveq2 5870	. . . . . . . . . 10 |- ( d = D -> (pmTrsp ` d ) = (pmTrsp ` D ) )
13	12	rneqd 5065	. . . . . . . . 9 |- ( d = D -> ran (pmTrsp ` d ) = ran (pmTrsp ` D ) )
14		psgnfval.t	. . . . . . . . 9 |- T = ran (pmTrsp ` D )
15	13, 14	syl6eqr 2505	. . . . . . . 8 |- ( d = D -> ran (pmTrsp ` d ) = T )
16		wrdeq 12696	. . . . . . . 8 |- ( ran (pmTrsp ` d ) = T -> Word ran (pmTrsp ` d ) = Word T )
17	15, 16	syl 17	. . . . . . 7 |- ( d = D -> Word ran (pmTrsp ` d ) = Word T )
18	4	oveq1d 6310	. . . . . . . . 9 |- ( d = D -> ( ( SymGrp ` d ) gsumg w ) = ( G gsumg w ) )
19	18	eqeq2d 2463	. . . . . . . 8 |- ( d = D -> ( x = ( ( SymGrp ` d ) gsumg w ) <-> x = ( G gsumg w ) ) )
20	19	anbi1d 712	. . . . . . 7 |- ( d = D -> ( ( x = ( ( SymGrp ` d ) gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
21	17, 20	rexeqbidv 3004	. . . . . 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 5570	. . . . 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 4484	. . . 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 17144	. . . 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 5880	. . . . . . 7 |- ( Base ` G ) e. _V
26	6, 25	eqeltri 2527	. . . . . 6 |- B e. _V
27	10, 26	rabex2 4559	. . . . 5 |- F e. _V
28	27	mptex 6141	. . . 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 5953	. . 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 5864	. . . 4 |- ( -. D e. _V -> (pmSgn ` D ) = (/) )
31		fvprc 5864	. . . . . . . . . . . . 13 |- ( -. D e. _V -> ( SymGrp ` D ) = (/) )
32	3, 31	syl5eq 2499	. . . . . . . . . . . 12 |- ( -. D e. _V -> G = (/) )
33	32	fveq2d 5874	. . . . . . . . . . 11 |- ( -. D e. _V -> ( Base ` G ) = ( Base ` (/) ) )
34		base0 15174	. . . . . . . . . . 11 |- (/) = ( Base ` (/) )
35	33, 34	syl6eqr 2505	. . . . . . . . . 10 |- ( -. D e. _V -> ( Base ` G ) = (/) )
36	6, 35	syl5eq 2499	. . . . . . . . 9 |- ( -. D e. _V -> B = (/) )
37		rabeq 3040	. . . . . . . . 9 |- ( B = (/) -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
38	36, 37	syl 17	. . . . . . . 8 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = { p e. (/) | dom ( p \ _I ) e. Fin } )
39		rab0 3755	. . . . . . . 8 |- { p e. (/) | dom ( p \ _I ) e. Fin } = (/)
40	38, 39	syl6eq 2503	. . . . . . 7 |- ( -. D e. _V -> { p e. B | dom ( p \ _I ) e. Fin } = (/) )
41	10, 40	syl5eq 2499	. . . . . 6 |- ( -. D e. _V -> F = (/) )
42	41	mpteq1d 4487	. . . . 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 5710	. . . . 5 |- ( x e. (/) |-> ( iota s E. w e. Word T ( x = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) = (/)
44	42, 43	syl6eq 2503	. . . 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 2490	. . 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 168	. 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 2475	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 371   = wceq 1446   e. wcel 1889   E.wrex 2740   {crab 2743   _Vcvv 3047   \ cdif 3403   (/)c0 3733   |-> cmpt 4464   _I cid 4747   dom cdm 4837   ran crn 4838   iotacio 5547   ` cfv 5585  (class class class)co 6295   Fincfn 7574   1c1 9545   -ucneg 9866   ^cexp 12279   #chash 12522  Word cword 12663   Basecbs 15133   gsum_g cgsu 15351   SymGrpcsymg 17030  pmTrspcpmtr 17094  pmSgncpsgn 17142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-slot 15137  df-base 15138  df-psgn 17144

This theorem is referenced by:  psgnfn  17154  psgnval  17160  psgnfvalfi  17166