Theorem cayleylem2 16038

Description: Lemma for cayley 16039. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
cayleylem1.x	|- X = ( Base ` G )
cayleylem1.p	|- .+ = ( +g ` G )
cayleylem1.u	|- .0. = ( 0g ` G )
cayleylem1.h	|- H = ( SymGrp ` X )
cayleylem1.s	|- S = ( Base ` H )
cayleylem1.f	|- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )

Assertion

Ref	Expression
cayleylem2	|- ( G e. Grp -> F : X -1-1-> S )

Distinct variable groups:   g, a, .+   G, a, g   g, H   X, a, g   .0. , a

Allowed substitution hints:   S( g, a)   F( g, a)   H( a)   .0. ( g)

Proof of Theorem cayleylem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5799	. . . 4 |- ( ( F ` x ) = ( 0g ` H ) -> ( ( F ` x ) ` .0. ) = ( ( 0g ` H ) ` .0. ) )
2		simpr 461	. . . . . . 7 |- ( ( G e. Grp /\ x e. X ) -> x e. X )
3		cayleylem1.x	. . . . . . . . 9 |- X = ( Base ` G )
4		cayleylem1.u	. . . . . . . . 9 |- .0. = ( 0g ` G )
5	3, 4	grpidcl 15686	. . . . . . . 8 |- ( G e. Grp -> .0. e. X )
6	5	adantr 465	. . . . . . 7 |- ( ( G e. Grp /\ x e. X ) -> .0. e. X )
7		cayleylem1.f	. . . . . . . 8 |- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )
8	7, 3	grplactval 15743	. . . . . . 7 |- ( ( x e. X /\ .0. e. X ) -> ( ( F ` x ) ` .0. ) = ( x .+ .0. ) )
9	2, 6, 8	syl2anc 661	. . . . . 6 |- ( ( G e. Grp /\ x e. X ) -> ( ( F ` x ) ` .0. ) = ( x .+ .0. ) )
10		cayleylem1.p	. . . . . . 7 |- .+ = ( +g ` G )
11	3, 10, 4	grprid 15689	. . . . . 6 |- ( ( G e. Grp /\ x e. X ) -> ( x .+ .0. ) = x )
12	9, 11	eqtrd 2495	. . . . 5 |- ( ( G e. Grp /\ x e. X ) -> ( ( F ` x ) ` .0. ) = x )
13		fvex 5810	. . . . . . . . 9 |- ( Base ` G ) e. _V
14	3, 13	eqeltri 2538	. . . . . . . 8 |- X e. _V
15		cayleylem1.h	. . . . . . . . 9 |- H = ( SymGrp ` X )
16	15	symgid 16026	. . . . . . . 8 |- ( X e. _V -> ( _I |` X ) = ( 0g ` H ) )
17	14, 16	ax-mp 5	. . . . . . 7 |- ( _I |` X ) = ( 0g ` H )
18	17	fveq1i 5801	. . . . . 6 |- ( ( _I |` X ) ` .0. ) = ( ( 0g ` H ) ` .0. )
19		fvresi 6014	. . . . . . 7 |- ( .0. e. X -> ( ( _I |` X ) ` .0. ) = .0. )
20	6, 19	syl 16	. . . . . 6 |- ( ( G e. Grp /\ x e. X ) -> ( ( _I |` X ) ` .0. ) = .0. )
21	18, 20	syl5eqr 2509	. . . . 5 |- ( ( G e. Grp /\ x e. X ) -> ( ( 0g ` H ) ` .0. ) = .0. )
22	12, 21	eqeq12d 2476	. . . 4 |- ( ( G e. Grp /\ x e. X ) -> ( ( ( F ` x ) ` .0. ) = ( ( 0g ` H ) ` .0. ) <-> x = .0. ) )
23	1, 22	syl5ib 219	. . 3 |- ( ( G e. Grp /\ x e. X ) -> ( ( F ` x ) = ( 0g ` H ) -> x = .0. ) )
24	23	ralrimiva 2830	. 2 |- ( G e. Grp -> A. x e. X ( ( F ` x ) = ( 0g ` H ) -> x = .0. ) )
25		cayleylem1.s	. . . 4 |- S = ( Base ` H )
26	3, 10, 4, 15, 25, 7	cayleylem1 16037	. . 3 |- ( G e. Grp -> F e. ( G GrpHom H ) )
27		eqid 2454	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
28	3, 25, 4, 27	ghmf1 15895	. . 3 |- ( F e. ( G GrpHom H ) -> ( F : X -1-1-> S <-> A. x e. X ( ( F ` x ) = ( 0g ` H ) -> x = .0. ) ) )
29	26, 28	syl 16	. 2 |- ( G e. Grp -> ( F : X -1-1-> S <-> A. x e. X ( ( F ` x ) = ( 0g ` H ) -> x = .0. ) ) )
30	24, 29	mpbird 232	1 |- ( G e. Grp -> F : X -1-1-> S )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2799   _Vcvv 3078   |-> cmpt 4459   _I cid 4740   |` cres 4951   -1-1->wf1 5524   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   0gc0g 14498   Grpcgrp 15530   GrpHom cghm 15864   SymGrpcsymg 16002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-tset 14377  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-ghm 15865  df-ga 15928  df-symg 16003

This theorem is referenced by:  cayley  16039