Theorem cayleylem2 16640

Description: Lemma for cayley 16641. (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 5847	. . . 4 |- ( ( F ` x ) = ( 0g ` H ) -> ( ( F ` x ) ` .0. ) = ( ( 0g ` H ) ` .0. ) )
2		simpr 459	. . . . . . 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 16280	. . . . . . . 8 |- ( G e. Grp -> .0. e. X )
6	5	adantr 463	. . . . . . 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 16339	. . . . . . 7 |- ( ( x e. X /\ .0. e. X ) -> ( ( F ` x ) ` .0. ) = ( x .+ .0. ) )
9	2, 6, 8	syl2anc 659	. . . . . 6 |- ( ( G e. Grp /\ x e. X ) -> ( ( F ` x ) ` .0. ) = ( x .+ .0. ) )
10		cayleylem1.p	. . . . . . 7 |- .+ = ( +g ` G )
11	3, 10, 4	grprid 16283	. . . . . 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 5858	. . . . . . . . 9 |- ( Base ` G ) e. _V
14	3, 13	eqeltri 2538	. . . . . . . 8 |- X e. _V
15		cayleylem1.h	. . . . . . . . 9 |- H = ( SymGrp ` X )
16	15	symgid 16628	. . . . . . . 8 |- ( X e. _V -> ( _I |` X ) = ( 0g ` H ) )
17	14, 16	ax-mp 5	. . . . . . 7 |- ( _I |` X ) = ( 0g ` H )
18	17	fveq1i 5849	. . . . . 6 |- ( ( _I |` X ) ` .0. ) = ( ( 0g ` H ) ` .0. )
19		fvresi 6073	. . . . . . 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 2868	. 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 16639	. . 3 |- ( G e. Grp -> F e. ( G GrpHom H ) )
27		eqid 2454	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
28	3, 25, 4, 27	ghmf1 16497	. . 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 367   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   |-> cmpt 4497   _I cid 4779   |` cres 4990   -1-1->wf1 5567   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Grpcgrp 16255   GrpHom cghm 16466   SymGrpcsymg 16604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-tset 14806  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-ghm 16467  df-ga 16530  df-symg 16605

This theorem is referenced by:  cayley  16641