Theorem gsmsymgrfix 17848

Description: The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.)

Hypotheses

Ref	Expression
gsmsymgrfix.s	|- S = ( SymGrp ` N )
gsmsymgrfix.b	|- B = ( Base ` S )

Assertion

Ref	Expression
gsmsymgrfix	|- ( ( N e. Fin /\ K e. N /\ W e. Word B ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) )

Distinct variable groups:   B, i   i, K   i, N   i, W

Allowed substitution hint:   S( i)

Proof of Theorem gsmsymgrfix

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . . . . 11 |- w e. _V
2		hasheq0 13154	. . . . . . . . . . 11 |- ( w e. _V -> ( ( # ` w ) = 0 <-> w = (/) ) )
3	1, 2	ax-mp 5	. . . . . . . . . 10 |- ( ( # ` w ) = 0 <-> w = (/) )
4	3	biimpri 218	. . . . . . . . 9 |- ( w = (/) -> ( # ` w ) = 0 )
5	4	oveq2d 6666	. . . . . . . 8 |- ( w = (/) -> ( 0..^ ( # ` w ) ) = ( 0..^ 0 ) )
6		fzo0 12492	. . . . . . . 8 |- ( 0..^ 0 ) = (/)
7	5, 6	syl6eq 2672	. . . . . . 7 |- ( w = (/) -> ( 0..^ ( # ` w ) ) = (/) )
8		fveq1 6190	. . . . . . . . 9 |- ( w = (/) -> ( w ` i ) = ( (/) ` i ) )
9	8	fveq1d 6193	. . . . . . . 8 |- ( w = (/) -> ( ( w ` i ) ` K ) = ( ( (/) ` i ) ` K ) )
10	9	eqeq1d 2624	. . . . . . 7 |- ( w = (/) -> ( ( ( w ` i ) ` K ) = K <-> ( ( (/) ` i ) ` K ) = K ) )
11	7, 10	raleqbidv 3152	. . . . . 6 |- ( w = (/) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K <-> A. i e. (/) ( ( (/) ` i ) ` K ) = K ) )
12		oveq2 6658	. . . . . . . 8 |- ( w = (/) -> ( S gsumg w ) = ( S gsumg (/) ) )
13	12	fveq1d 6193	. . . . . . 7 |- ( w = (/) -> ( ( S gsumg w ) ` K ) = ( ( S gsumg (/) ) ` K ) )
14	13	eqeq1d 2624	. . . . . 6 |- ( w = (/) -> ( ( ( S gsumg w ) ` K ) = K <-> ( ( S gsumg (/) ) ` K ) = K ) )
15	11, 14	imbi12d 334	. . . . 5 |- ( w = (/) -> ( ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) <-> ( A. i e. (/) ( ( (/) ` i ) ` K ) = K -> ( ( S gsumg (/) ) ` K ) = K ) ) )
16	15	imbi2d 330	. . . 4 |- ( w = (/) -> ( ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) ) <-> ( ( N e. Fin /\ K e. N ) -> ( A. i e. (/) ( ( (/) ` i ) ` K ) = K -> ( ( S gsumg (/) ) ` K ) = K ) ) ) )
17		fveq2 6191	. . . . . . . 8 |- ( w = y -> ( # ` w ) = ( # ` y ) )
18	17	oveq2d 6666	. . . . . . 7 |- ( w = y -> ( 0..^ ( # ` w ) ) = ( 0..^ ( # ` y ) ) )
19		fveq1 6190	. . . . . . . . 9 |- ( w = y -> ( w ` i ) = ( y ` i ) )
20	19	fveq1d 6193	. . . . . . . 8 |- ( w = y -> ( ( w ` i ) ` K ) = ( ( y ` i ) ` K ) )
21	20	eqeq1d 2624	. . . . . . 7 |- ( w = y -> ( ( ( w ` i ) ` K ) = K <-> ( ( y ` i ) ` K ) = K ) )
22	18, 21	raleqbidv 3152	. . . . . 6 |- ( w = y -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K <-> A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K ) )
23		oveq2 6658	. . . . . . . 8 |- ( w = y -> ( S gsumg w ) = ( S gsumg y ) )
24	23	fveq1d 6193	. . . . . . 7 |- ( w = y -> ( ( S gsumg w ) ` K ) = ( ( S gsumg y ) ` K ) )
25	24	eqeq1d 2624	. . . . . 6 |- ( w = y -> ( ( ( S gsumg w ) ` K ) = K <-> ( ( S gsumg y ) ` K ) = K ) )
26	22, 25	imbi12d 334	. . . . 5 |- ( w = y -> ( ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) <-> ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) )
27	26	imbi2d 330	. . . 4 |- ( w = y -> ( ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) ) <-> ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) ) )
28		fveq2 6191	. . . . . . . 8 |- ( w = ( y ++ <" z "> ) -> ( # ` w ) = ( # ` ( y ++ <" z "> ) ) )
29	28	oveq2d 6666	. . . . . . 7 |- ( w = ( y ++ <" z "> ) -> ( 0..^ ( # ` w ) ) = ( 0..^ ( # ` ( y ++ <" z "> ) ) ) )
30		fveq1 6190	. . . . . . . . 9 |- ( w = ( y ++ <" z "> ) -> ( w ` i ) = ( ( y ++ <" z "> ) ` i ) )
31	30	fveq1d 6193	. . . . . . . 8 |- ( w = ( y ++ <" z "> ) -> ( ( w ` i ) ` K ) = ( ( ( y ++ <" z "> ) ` i ) ` K ) )
32	31	eqeq1d 2624	. . . . . . 7 |- ( w = ( y ++ <" z "> ) -> ( ( ( w ` i ) ` K ) = K <-> ( ( ( y ++ <" z "> ) ` i ) ` K ) = K ) )
33	29, 32	raleqbidv 3152	. . . . . 6 |- ( w = ( y ++ <" z "> ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K <-> A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K ) )
34		oveq2 6658	. . . . . . . 8 |- ( w = ( y ++ <" z "> ) -> ( S gsumg w ) = ( S gsumg ( y ++ <" z "> ) ) )
35	34	fveq1d 6193	. . . . . . 7 |- ( w = ( y ++ <" z "> ) -> ( ( S gsumg w ) ` K ) = ( ( S gsumg ( y ++ <" z "> ) ) ` K ) )
36	35	eqeq1d 2624	. . . . . 6 |- ( w = ( y ++ <" z "> ) -> ( ( ( S gsumg w ) ` K ) = K <-> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) )
37	33, 36	imbi12d 334	. . . . 5 |- ( w = ( y ++ <" z "> ) -> ( ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) <-> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) ) )
38	37	imbi2d 330	. . . 4 |- ( w = ( y ++ <" z "> ) -> ( ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) ) <-> ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) ) ) )
39		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( # ` w ) = ( # ` W ) )
40	39	oveq2d 6666	. . . . . . 7 |- ( w = W -> ( 0..^ ( # ` w ) ) = ( 0..^ ( # ` W ) ) )
41		fveq1 6190	. . . . . . . . 9 |- ( w = W -> ( w ` i ) = ( W ` i ) )
42	41	fveq1d 6193	. . . . . . . 8 |- ( w = W -> ( ( w ` i ) ` K ) = ( ( W ` i ) ` K ) )
43	42	eqeq1d 2624	. . . . . . 7 |- ( w = W -> ( ( ( w ` i ) ` K ) = K <-> ( ( W ` i ) ` K ) = K ) )
44	40, 43	raleqbidv 3152	. . . . . 6 |- ( w = W -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K <-> A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K ) )
45		oveq2 6658	. . . . . . . 8 |- ( w = W -> ( S gsumg w ) = ( S gsumg W ) )
46	45	fveq1d 6193	. . . . . . 7 |- ( w = W -> ( ( S gsumg w ) ` K ) = ( ( S gsumg W ) ` K ) )
47	46	eqeq1d 2624	. . . . . 6 |- ( w = W -> ( ( ( S gsumg w ) ` K ) = K <-> ( ( S gsumg W ) ` K ) = K ) )
48	44, 47	imbi12d 334	. . . . 5 |- ( w = W -> ( ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) <-> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) ) )
49	48	imbi2d 330	. . . 4 |- ( w = W -> ( ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` w ) ) ( ( w ` i ) ` K ) = K -> ( ( S gsumg w ) ` K ) = K ) ) <-> ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) ) ) )
50		gsmsymgrfix.s	. . . . . . . . . 10 |- S = ( SymGrp ` N )
51	50	symgid 17821	. . . . . . . . 9 |- ( N e. Fin -> ( _I |` N ) = ( 0g ` S ) )
52	51	adantr 481	. . . . . . . 8 |- ( ( N e. Fin /\ K e. N ) -> ( _I |` N ) = ( 0g ` S ) )
53		eqid 2622	. . . . . . . . 9 |- ( 0g ` S ) = ( 0g ` S )
54	53	gsum0 17278	. . . . . . . 8 |- ( S gsumg (/) ) = ( 0g ` S )
55	52, 54	syl6reqr 2675	. . . . . . 7 |- ( ( N e. Fin /\ K e. N ) -> ( S gsumg (/) ) = ( _I |` N ) )
56	55	fveq1d 6193	. . . . . 6 |- ( ( N e. Fin /\ K e. N ) -> ( ( S gsumg (/) ) ` K ) = ( ( _I |` N ) ` K ) )
57		fvresi 6439	. . . . . . 7 |- ( K e. N -> ( ( _I |` N ) ` K ) = K )
58	57	adantl 482	. . . . . 6 |- ( ( N e. Fin /\ K e. N ) -> ( ( _I |` N ) ` K ) = K )
59	56, 58	eqtrd 2656	. . . . 5 |- ( ( N e. Fin /\ K e. N ) -> ( ( S gsumg (/) ) ` K ) = K )
60	59	a1d 25	. . . 4 |- ( ( N e. Fin /\ K e. N ) -> ( A. i e. (/) ( ( (/) ` i ) ` K ) = K -> ( ( S gsumg (/) ) ` K ) = K ) )
61		ccatws1len 13398	. . . . . . . . . 10 |- ( ( y e. Word B /\ z e. B ) -> ( # ` ( y ++ <" z "> ) ) = ( ( # ` y ) + 1 ) )
62	61	oveq2d 6666	. . . . . . . . 9 |- ( ( y e. Word B /\ z e. B ) -> ( 0..^ ( # ` ( y ++ <" z "> ) ) ) = ( 0..^ ( ( # ` y ) + 1 ) ) )
63	62	raleqdv 3144	. . . . . . . 8 |- ( ( y e. Word B /\ z e. B ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K <-> A. i e. ( 0..^ ( ( # ` y ) + 1 ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K ) )
64	63	adantr 481	. . . . . . 7 |- ( ( ( y e. Word B /\ z e. B ) /\ ( ( N e. Fin /\ K e. N ) /\ ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K <-> A. i e. ( 0..^ ( ( # ` y ) + 1 ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K ) )
65		gsmsymgrfix.b	. . . . . . . . 9 |- B = ( Base ` S )
66	50, 65	gsmsymgrfixlem1 17847	. . . . . . . 8 |- ( ( ( y e. Word B /\ z e. B ) /\ ( N e. Fin /\ K e. N ) /\ ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) -> ( A. i e. ( 0..^ ( ( # ` y ) + 1 ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) )
67	66	3expb 1266	. . . . . . 7 |- ( ( ( y e. Word B /\ z e. B ) /\ ( ( N e. Fin /\ K e. N ) /\ ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) ) -> ( A. i e. ( 0..^ ( ( # ` y ) + 1 ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) )
68	64, 67	sylbid 230	. . . . . 6 |- ( ( ( y e. Word B /\ z e. B ) /\ ( ( N e. Fin /\ K e. N ) /\ ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) )
69	68	exp32 631	. . . . 5 |- ( ( y e. Word B /\ z e. B ) -> ( ( N e. Fin /\ K e. N ) -> ( ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) ) ) )
70	69	a2d 29	. . . 4 |- ( ( y e. Word B /\ z e. B ) -> ( ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` y ) ) ( ( y ` i ) ` K ) = K -> ( ( S gsumg y ) ` K ) = K ) ) -> ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` ( y ++ <" z "> ) ) ) ( ( ( y ++ <" z "> ) ` i ) ` K ) = K -> ( ( S gsumg ( y ++ <" z "> ) ) ` K ) = K ) ) ) )
71	16, 27, 38, 49, 60, 70	wrdind 13476	. . 3 |- ( W e. Word B -> ( ( N e. Fin /\ K e. N ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) ) )
72	71	com12 32	. 2 |- ( ( N e. Fin /\ K e. N ) -> ( W e. Word B -> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) ) )
73	72	3impia 1261	1 |- ( ( N e. Fin /\ K e. N /\ W e. Word B ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( ( W ` i ) ` K ) = K -> ( ( S gsumg W ) ` K ) = K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   _I cid 5023   |` cres 5116   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   SymGrpcsymg 17797

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-symg 17798

This theorem is referenced by:  psgndiflemB  19946