Theorem sylow2alem1 18032

Description: Lemma for sylow2a 18034. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypotheses

Ref	Expression
sylow2a.x	|- X = ( Base ` G )
sylow2a.m	|- ( ph -> .(+) e. ( G GrpAct Y ) )
sylow2a.p	|- ( ph -> P pGrp G )
sylow2a.f	|- ( ph -> X e. Fin )
sylow2a.y	|- ( ph -> Y e. Fin )
sylow2a.z	|- Z = { u e. Y | A. h e. X ( h .(+) u ) = u }
sylow2a.r	|- .~ = { <. x , y >. | ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) }

Assertion

Ref	Expression
sylow2alem1	|- ( ( ph /\ A e. Z ) -> [ A ] .~ = { A } )

Distinct variable groups:   .~ , h   g, h, u, x, y, A   g, G, x, y   .(+) , g, h, u, x, y   g, X, h, u, x, y   ph, h   g, Y, h, u, x, y

Allowed substitution hints:   ph( x, y, u, g)   P( x, y, u, g, h)   .~ ( x, y, u, g)   G( u, h)   Z( x, y, u, g, h)

Proof of Theorem sylow2alem1

Dummy variables k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- w e. _V
2		simpr 477	. . . . . 6 |- ( ( ph /\ A e. Z ) -> A e. Z )
3		elecg 7785	. . . . . 6 |- ( ( w e. _V /\ A e. Z ) -> ( w e. [ A ] .~ <-> A .~ w ) )
4	1, 2, 3	sylancr 695	. . . . 5 |- ( ( ph /\ A e. Z ) -> ( w e. [ A ] .~ <-> A .~ w ) )
5		sylow2a.r	. . . . . . . 8 |- .~ = { <. x , y >. | ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) }
6	5	gaorb 17740	. . . . . . 7 |- ( A .~ w <-> ( A e. Y /\ w e. Y /\ E. k e. X ( k .(+) A ) = w ) )
7	6	simp3bi 1078	. . . . . 6 |- ( A .~ w -> E. k e. X ( k .(+) A ) = w )
8		oveq2 6658	. . . . . . . . . . . . . 14 |- ( u = A -> ( h .(+) u ) = ( h .(+) A ) )
9		id 22	. . . . . . . . . . . . . 14 |- ( u = A -> u = A )
10	8, 9	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( u = A -> ( ( h .(+) u ) = u <-> ( h .(+) A ) = A ) )
11	10	ralbidv 2986	. . . . . . . . . . . 12 |- ( u = A -> ( A. h e. X ( h .(+) u ) = u <-> A. h e. X ( h .(+) A ) = A ) )
12		sylow2a.z	. . . . . . . . . . . 12 |- Z = { u e. Y | A. h e. X ( h .(+) u ) = u }
13	11, 12	elrab2 3366	. . . . . . . . . . 11 |- ( A e. Z <-> ( A e. Y /\ A. h e. X ( h .(+) A ) = A ) )
14	2, 13	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ A e. Z ) -> ( A e. Y /\ A. h e. X ( h .(+) A ) = A ) )
15	14	simprd 479	. . . . . . . . 9 |- ( ( ph /\ A e. Z ) -> A. h e. X ( h .(+) A ) = A )
16		oveq1 6657	. . . . . . . . . . 11 |- ( h = k -> ( h .(+) A ) = ( k .(+) A ) )
17	16	eqeq1d 2624	. . . . . . . . . 10 |- ( h = k -> ( ( h .(+) A ) = A <-> ( k .(+) A ) = A ) )
18	17	rspccva 3308	. . . . . . . . 9 |- ( ( A. h e. X ( h .(+) A ) = A /\ k e. X ) -> ( k .(+) A ) = A )
19	15, 18	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ A e. Z ) /\ k e. X ) -> ( k .(+) A ) = A )
20		eqeq1 2626	. . . . . . . 8 |- ( ( k .(+) A ) = w -> ( ( k .(+) A ) = A <-> w = A ) )
21	19, 20	syl5ibcom 235	. . . . . . 7 |- ( ( ( ph /\ A e. Z ) /\ k e. X ) -> ( ( k .(+) A ) = w -> w = A ) )
22	21	rexlimdva 3031	. . . . . 6 |- ( ( ph /\ A e. Z ) -> ( E. k e. X ( k .(+) A ) = w -> w = A ) )
23	7, 22	syl5 34	. . . . 5 |- ( ( ph /\ A e. Z ) -> ( A .~ w -> w = A ) )
24	4, 23	sylbid 230	. . . 4 |- ( ( ph /\ A e. Z ) -> ( w e. [ A ] .~ -> w = A ) )
25		velsn 4193	. . . 4 |- ( w e. { A } <-> w = A )
26	24, 25	syl6ibr 242	. . 3 |- ( ( ph /\ A e. Z ) -> ( w e. [ A ] .~ -> w e. { A } ) )
27	26	ssrdv 3609	. 2 |- ( ( ph /\ A e. Z ) -> [ A ] .~ C_ { A } )
28		sylow2a.m	. . . . . . 7 |- ( ph -> .(+) e. ( G GrpAct Y ) )
29		sylow2a.x	. . . . . . . 8 |- X = ( Base ` G )
30	5, 29	gaorber 17741	. . . . . . 7 |- ( .(+) e. ( G GrpAct Y ) -> .~ Er Y )
31	28, 30	syl 17	. . . . . 6 |- ( ph -> .~ Er Y )
32	31	adantr 481	. . . . 5 |- ( ( ph /\ A e. Z ) -> .~ Er Y )
33	14	simpld 475	. . . . 5 |- ( ( ph /\ A e. Z ) -> A e. Y )
34	32, 33	erref 7762	. . . 4 |- ( ( ph /\ A e. Z ) -> A .~ A )
35		elecg 7785	. . . . 5 |- ( ( A e. Z /\ A e. Z ) -> ( A e. [ A ] .~ <-> A .~ A ) )
36	2, 35	sylancom 701	. . . 4 |- ( ( ph /\ A e. Z ) -> ( A e. [ A ] .~ <-> A .~ A ) )
37	34, 36	mpbird 247	. . 3 |- ( ( ph /\ A e. Z ) -> A e. [ A ] .~ )
38	37	snssd 4340	. 2 |- ( ( ph /\ A e. Z ) -> { A } C_ [ A ] .~ )
39	27, 38	eqssd 3620	1 |- ( ( ph /\ A e. Z ) -> [ A ] .~ = { A } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   Fincfn 7955   Basecbs 15857   GrpAct cga 17722   pGrp cpgp 17946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-ec 7744  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ga 17723

This theorem is referenced by:  sylow2alem2  18033  sylow2a  18034