Theorem fvcosymgeq 17849

Description: The values of two compositions of permutations are equal if the values of the composed permutations are pairwise equal. (Contributed by AV, 26-Jan-2019.)

Hypotheses

Ref	Expression
gsmsymgrfix.s	|- S = ( SymGrp ` N )
gsmsymgrfix.b	|- B = ( Base ` S )
gsmsymgreq.z	|- Z = ( SymGrp ` M )
gsmsymgreq.p	|- P = ( Base ` Z )
gsmsymgreq.i	|- I = ( N i^i M )

Assertion

Ref	Expression
fvcosymgeq	|- ( ( G e. B /\ K e. P ) -> ( ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )

Distinct variable groups:   n, F   n, G   n, H   n, I   n, K   n, X

Allowed substitution hints:   B( n)   P( n)   S( n)   M( n)   N( n)   Z( n)

Proof of Theorem fvcosymgeq

Step	Hyp	Ref	Expression
1		gsmsymgrfix.s	. . . . . . 7 |- S = ( SymGrp ` N )
2		gsmsymgrfix.b	. . . . . . 7 |- B = ( Base ` S )
3	1, 2	symgbasf 17804	. . . . . 6 |- ( G e. B -> G : N --> N )
4		ffn 6045	. . . . . 6 |- ( G : N --> N -> G Fn N )
5	3, 4	syl 17	. . . . 5 |- ( G e. B -> G Fn N )
6		gsmsymgreq.z	. . . . . . 7 |- Z = ( SymGrp ` M )
7		gsmsymgreq.p	. . . . . . 7 |- P = ( Base ` Z )
8	6, 7	symgbasf 17804	. . . . . 6 |- ( K e. P -> K : M --> M )
9		ffn 6045	. . . . . 6 |- ( K : M --> M -> K Fn M )
10	8, 9	syl 17	. . . . 5 |- ( K e. P -> K Fn M )
11	5, 10	anim12i 590	. . . 4 |- ( ( G e. B /\ K e. P ) -> ( G Fn N /\ K Fn M ) )
12	11	adantr 481	. . 3 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> ( G Fn N /\ K Fn M ) )
13		gsmsymgreq.i	. . . . . . . 8 |- I = ( N i^i M )
14	13	eleq2i 2693	. . . . . . 7 |- ( X e. I <-> X e. ( N i^i M ) )
15	14	biimpi 206	. . . . . 6 |- ( X e. I -> X e. ( N i^i M ) )
16	15	3ad2ant1 1082	. . . . 5 |- ( ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) -> X e. ( N i^i M ) )
17	16	adantl 482	. . . 4 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> X e. ( N i^i M ) )
18		simpr2 1068	. . . 4 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> ( G ` X ) = ( K ` X ) )
19	1, 2	symgbasf1o 17803	. . . . . . . . . . 11 |- ( G e. B -> G : N -1-1-onto-> N )
20		dff1o5 6146	. . . . . . . . . . . 12 |- ( G : N -1-1-onto-> N <-> ( G : N -1-1-> N /\ ran G = N ) )
21		eqcom 2629	. . . . . . . . . . . . . 14 |- ( ran G = N <-> N = ran G )
22	21	biimpi 206	. . . . . . . . . . . . 13 |- ( ran G = N -> N = ran G )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( G : N -1-1-> N /\ ran G = N ) -> N = ran G )
24	20, 23	sylbi 207	. . . . . . . . . . 11 |- ( G : N -1-1-onto-> N -> N = ran G )
25	19, 24	syl 17	. . . . . . . . . 10 |- ( G e. B -> N = ran G )
26	6, 7	symgbasf1o 17803	. . . . . . . . . . 11 |- ( K e. P -> K : M -1-1-onto-> M )
27		dff1o5 6146	. . . . . . . . . . . 12 |- ( K : M -1-1-onto-> M <-> ( K : M -1-1-> M /\ ran K = M ) )
28		eqcom 2629	. . . . . . . . . . . . . 14 |- ( ran K = M <-> M = ran K )
29	28	biimpi 206	. . . . . . . . . . . . 13 |- ( ran K = M -> M = ran K )
30	29	adantl 482	. . . . . . . . . . . 12 |- ( ( K : M -1-1-> M /\ ran K = M ) -> M = ran K )
31	27, 30	sylbi 207	. . . . . . . . . . 11 |- ( K : M -1-1-onto-> M -> M = ran K )
32	26, 31	syl 17	. . . . . . . . . 10 |- ( K e. P -> M = ran K )
33	25, 32	ineqan12d 3816	. . . . . . . . 9 |- ( ( G e. B /\ K e. P ) -> ( N i^i M ) = ( ran G i^i ran K ) )
34	13, 33	syl5eq 2668	. . . . . . . 8 |- ( ( G e. B /\ K e. P ) -> I = ( ran G i^i ran K ) )
35	34	raleqdv 3144	. . . . . . 7 |- ( ( G e. B /\ K e. P ) -> ( A. n e. I ( F ` n ) = ( H ` n ) <-> A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) ) )
36	35	biimpcd 239	. . . . . 6 |- ( A. n e. I ( F ` n ) = ( H ` n ) -> ( ( G e. B /\ K e. P ) -> A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) ) )
37	36	3ad2ant3 1084	. . . . 5 |- ( ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) -> ( ( G e. B /\ K e. P ) -> A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) ) )
38	37	impcom 446	. . . 4 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) )
39	17, 18, 38	3jca 1242	. . 3 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> ( X e. ( N i^i M ) /\ ( G ` X ) = ( K ` X ) /\ A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) ) )
40		fvcofneq 6367	. . 3 |- ( ( G Fn N /\ K Fn M ) -> ( ( X e. ( N i^i M ) /\ ( G ` X ) = ( K ` X ) /\ A. n e. ( ran G i^i ran K ) ( F ` n ) = ( H ` n ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )
41	12, 39, 40	sylc 65	. 2 |- ( ( ( G e. B /\ K e. P ) /\ ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) )
42	41	ex 450	1 |- ( ( G e. B /\ K e. P ) -> ( ( X e. I /\ ( G ` X ) = ( K ` X ) /\ A. n e. I ( F ` n ) = ( H ` n ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   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-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-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-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-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798

This theorem is referenced by:  gsmsymgreqlem1  17850