Theorem pmtr3ncom 17895

Description: Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)

Hypothesis

Ref	Expression
pmtr3ncom.t	|- T = (pmTrsp ` D )

Assertion

Ref	Expression
pmtr3ncom	|- ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) )

Distinct variable groups:   D, f, g   T, f, g

Allowed substitution hints:   V( f, g)

Proof of Theorem pmtr3ncom

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

Step	Hyp	Ref	Expression
1		hashge3el3dif 13268	. 2 |- ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) )
2		simprl 794	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> D e. V )
3		prssi 4353	. . . . . . . . 9 |- ( ( x e. D /\ y e. D ) -> { x , y } C_ D )
4	3	adantr 481	. . . . . . . 8 |- ( ( ( x e. D /\ y e. D ) /\ z e. D ) -> { x , y } C_ D )
5	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> { x , y } C_ D )
6		simplll 798	. . . . . . . . 9 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x e. D )
7		simplr 792	. . . . . . . . . 10 |- ( ( ( x e. D /\ y e. D ) /\ z e. D ) -> y e. D )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y e. D )
9		simpr1 1067	. . . . . . . . 9 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> x =/= y )
10		pr2nelem 8827	. . . . . . . . 9 |- ( ( x e. D /\ y e. D /\ x =/= y ) -> { x , y } ~~ 2o )
11	6, 8, 9, 10	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> { x , y } ~~ 2o )
12	11	adantr 481	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> { x , y } ~~ 2o )
13		pmtr3ncom.t	. . . . . . . 8 |- T = (pmTrsp ` D )
14		eqid 2622	. . . . . . . 8 |- ran T = ran T
15	13, 14	pmtrrn 17877	. . . . . . 7 |- ( ( D e. V /\ { x , y } C_ D /\ { x , y } ~~ 2o ) -> ( T ` { x , y } ) e. ran T )
16	2, 5, 12, 15	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> ( T ` { x , y } ) e. ran T )
17		prssi 4353	. . . . . . . . 9 |- ( ( y e. D /\ z e. D ) -> { y , z } C_ D )
18	17	adantll 750	. . . . . . . 8 |- ( ( ( x e. D /\ y e. D ) /\ z e. D ) -> { y , z } C_ D )
19	18	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> { y , z } C_ D )
20		simplr 792	. . . . . . . . 9 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> z e. D )
21		simpr3 1069	. . . . . . . . 9 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> y =/= z )
22		pr2nelem 8827	. . . . . . . . 9 |- ( ( y e. D /\ z e. D /\ y =/= z ) -> { y , z } ~~ 2o )
23	8, 20, 21, 22	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> { y , z } ~~ 2o )
24	23	adantr 481	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> { y , z } ~~ 2o )
25	13, 14	pmtrrn 17877	. . . . . . 7 |- ( ( D e. V /\ { y , z } C_ D /\ { y , z } ~~ 2o ) -> ( T ` { y , z } ) e. ran T )
26	2, 19, 24, 25	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> ( T ` { y , z } ) e. ran T )
27		df-3an 1039	. . . . . . . . 9 |- ( ( x e. D /\ y e. D /\ z e. D ) <-> ( ( x e. D /\ y e. D ) /\ z e. D ) )
28	27	biimpri 218	. . . . . . . 8 |- ( ( ( x e. D /\ y e. D ) /\ z e. D ) -> ( x e. D /\ y e. D /\ z e. D ) )
29	28	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> ( x e. D /\ y e. D /\ z e. D ) )
30		simplr 792	. . . . . . 7 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> ( x =/= y /\ x =/= z /\ y =/= z ) )
31		eqid 2622	. . . . . . . 8 |- ( T ` { x , y } ) = ( T ` { x , y } )
32		eqid 2622	. . . . . . . 8 |- ( T ` { y , z } ) = ( T ` { y , z } )
33	13, 31, 32	pmtr3ncomlem2 17894	. . . . . . 7 |- ( ( D e. V /\ ( x e. D /\ y e. D /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) -> ( ( T ` { y , z } ) o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. ( T ` { y , z } ) ) )
34	2, 29, 30, 33	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> ( ( T ` { y , z } ) o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. ( T ` { y , z } ) ) )
35		coeq2 5280	. . . . . . . 8 |- ( f = ( T ` { x , y } ) -> ( g o. f ) = ( g o. ( T ` { x , y } ) ) )
36		coeq1 5279	. . . . . . . 8 |- ( f = ( T ` { x , y } ) -> ( f o. g ) = ( ( T ` { x , y } ) o. g ) )
37	35, 36	neeq12d 2855	. . . . . . 7 |- ( f = ( T ` { x , y } ) -> ( ( g o. f ) =/= ( f o. g ) <-> ( g o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. g ) ) )
38		coeq1 5279	. . . . . . . 8 |- ( g = ( T ` { y , z } ) -> ( g o. ( T ` { x , y } ) ) = ( ( T ` { y , z } ) o. ( T ` { x , y } ) ) )
39		coeq2 5280	. . . . . . . 8 |- ( g = ( T ` { y , z } ) -> ( ( T ` { x , y } ) o. g ) = ( ( T ` { x , y } ) o. ( T ` { y , z } ) ) )
40	38, 39	neeq12d 2855	. . . . . . 7 |- ( g = ( T ` { y , z } ) -> ( ( g o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. g ) <-> ( ( T ` { y , z } ) o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. ( T ` { y , z } ) ) ) )
41	37, 40	rspc2ev 3324	. . . . . 6 |- ( ( ( T ` { x , y } ) e. ran T /\ ( T ` { y , z } ) e. ran T /\ ( ( T ` { y , z } ) o. ( T ` { x , y } ) ) =/= ( ( T ` { x , y } ) o. ( T ` { y , z } ) ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) )
42	16, 26, 34, 41	syl3anc 1326	. . . . 5 |- ( ( ( ( ( x e. D /\ y e. D ) /\ z e. D ) /\ ( x =/= y /\ x =/= z /\ y =/= z ) ) /\ ( D e. V /\ 3 <_ ( # ` D ) ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) )
43	42	exp31 630	. . . 4 |- ( ( ( x e. D /\ y e. D ) /\ z e. D ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) -> ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) ) ) )
44	43	rexlimdva 3031	. . 3 |- ( ( x e. D /\ y e. D ) -> ( E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) -> ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) ) ) )
45	44	rexlimivv 3036	. 2 |- ( E. x e. D E. y e. D E. z e. D ( x =/= y /\ x =/= z /\ y =/= z ) -> ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) ) )
46	1, 45	mpcom 38	1 |- ( ( D e. V /\ 3 <_ ( # ` D ) ) -> E. f e. ran T E. g e. ran T ( g o. f ) =/= ( f o. g ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   {cpr 4179   class class class wbr 4653   ran crn 5115   o. ccom 5118   ` cfv 5888   2oc2o 7554   ~~ cen 7952   <_ cle 10075   3c3 11071   #chash 13117  pmTrspcpmtr 17861

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-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-pmtr 17862

This theorem is referenced by:  pgrpgt2nabl  42147