Theorem sgrp2nmndlem5 17416

Description: Lemma 5 for sgrp2nmnd 17417: M is not a monoid. (Contributed by AV, 29-Jan-2020.)

Hypotheses

Ref	Expression
mgm2nsgrp.s	|- S = { A , B }
mgm2nsgrp.b	|- ( Base ` M ) = S
sgrp2nmnd.o	|- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )

Assertion

Ref	Expression
sgrp2nmndlem5	|- ( ( # ` S ) = 2 -> M e/ Mnd )

Distinct variable groups:   x, S, y   x, A, y   x, B, y   x, M, y

Proof of Theorem sgrp2nmndlem5

Step	Hyp	Ref	Expression
1		mgm2nsgrp.s	. . 3 |- S = { A , B }
2	1	hashprdifel 13186	. 2 |- ( ( # ` S ) = 2 -> ( A e. S /\ B e. S /\ A =/= B ) )
3		mgm2nsgrp.b	. . . . . . . 8 |- ( Base ` M ) = S
4		sgrp2nmnd.o	. . . . . . . 8 |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )
5		eqid 2622	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
6	1, 3, 4, 5	sgrp2nmndlem2 17411	. . . . . . 7 |- ( ( A e. S /\ B e. S ) -> ( A ( +g ` M ) B ) = A )
7	6	3adant3 1081	. . . . . 6 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( A ( +g ` M ) B ) = A )
8		simp3 1063	. . . . . 6 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> A =/= B )
9	7, 8	eqnetrd 2861	. . . . 5 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( A ( +g ` M ) B ) =/= B )
10	9	olcd 408	. . . 4 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( ( A ( +g ` M ) A ) =/= A \/ ( A ( +g ` M ) B ) =/= B ) )
11		oveq2 6658	. . . . . . 7 |- ( y = A -> ( A ( +g ` M ) y ) = ( A ( +g ` M ) A ) )
12		id 22	. . . . . . 7 |- ( y = A -> y = A )
13	11, 12	neeq12d 2855	. . . . . 6 |- ( y = A -> ( ( A ( +g ` M ) y ) =/= y <-> ( A ( +g ` M ) A ) =/= A ) )
14		oveq2 6658	. . . . . . 7 |- ( y = B -> ( A ( +g ` M ) y ) = ( A ( +g ` M ) B ) )
15		id 22	. . . . . . 7 |- ( y = B -> y = B )
16	14, 15	neeq12d 2855	. . . . . 6 |- ( y = B -> ( ( A ( +g ` M ) y ) =/= y <-> ( A ( +g ` M ) B ) =/= B ) )
17	13, 16	rexprg 4235	. . . . 5 |- ( ( A e. S /\ B e. S ) -> ( E. y e. { A , B } ( A ( +g ` M ) y ) =/= y <-> ( ( A ( +g ` M ) A ) =/= A \/ ( A ( +g ` M ) B ) =/= B ) ) )
18	17	3adant3 1081	. . . 4 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( E. y e. { A , B } ( A ( +g ` M ) y ) =/= y <-> ( ( A ( +g ` M ) A ) =/= A \/ ( A ( +g ` M ) B ) =/= B ) ) )
19	10, 18	mpbird 247	. . 3 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> E. y e. { A , B } ( A ( +g ` M ) y ) =/= y )
20	1, 3, 4, 5	sgrp2nmndlem3 17412	. . . . . . 7 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( B ( +g ` M ) A ) = B )
21		necom 2847	. . . . . . . . . . 11 |- ( A =/= B <-> B =/= A )
22		df-ne 2795	. . . . . . . . . . 11 |- ( B =/= A <-> -. B = A )
23	21, 22	sylbb 209	. . . . . . . . . 10 |- ( A =/= B -> -. B = A )
24	23	3ad2ant3 1084	. . . . . . . . 9 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> -. B = A )
25	24	adantr 481	. . . . . . . 8 |- ( ( ( A e. S /\ B e. S /\ A =/= B ) /\ ( B ( +g ` M ) A ) = B ) -> -. B = A )
26		eqeq1 2626	. . . . . . . . 9 |- ( ( B ( +g ` M ) A ) = B -> ( ( B ( +g ` M ) A ) = A <-> B = A ) )
27	26	adantl 482	. . . . . . . 8 |- ( ( ( A e. S /\ B e. S /\ A =/= B ) /\ ( B ( +g ` M ) A ) = B ) -> ( ( B ( +g ` M ) A ) = A <-> B = A ) )
28	25, 27	mtbird 315	. . . . . . 7 |- ( ( ( A e. S /\ B e. S /\ A =/= B ) /\ ( B ( +g ` M ) A ) = B ) -> -. ( B ( +g ` M ) A ) = A )
29	20, 28	mpdan 702	. . . . . 6 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> -. ( B ( +g ` M ) A ) = A )
30	29	neqned 2801	. . . . 5 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( B ( +g ` M ) A ) =/= A )
31	30	orcd 407	. . . 4 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( ( B ( +g ` M ) A ) =/= A \/ ( B ( +g ` M ) B ) =/= B ) )
32		oveq2 6658	. . . . . . 7 |- ( y = A -> ( B ( +g ` M ) y ) = ( B ( +g ` M ) A ) )
33	32, 12	neeq12d 2855	. . . . . 6 |- ( y = A -> ( ( B ( +g ` M ) y ) =/= y <-> ( B ( +g ` M ) A ) =/= A ) )
34		oveq2 6658	. . . . . . 7 |- ( y = B -> ( B ( +g ` M ) y ) = ( B ( +g ` M ) B ) )
35	34, 15	neeq12d 2855	. . . . . 6 |- ( y = B -> ( ( B ( +g ` M ) y ) =/= y <-> ( B ( +g ` M ) B ) =/= B ) )
36	33, 35	rexprg 4235	. . . . 5 |- ( ( A e. S /\ B e. S ) -> ( E. y e. { A , B } ( B ( +g ` M ) y ) =/= y <-> ( ( B ( +g ` M ) A ) =/= A \/ ( B ( +g ` M ) B ) =/= B ) ) )
37	36	3adant3 1081	. . . 4 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( E. y e. { A , B } ( B ( +g ` M ) y ) =/= y <-> ( ( B ( +g ` M ) A ) =/= A \/ ( B ( +g ` M ) B ) =/= B ) ) )
38	31, 37	mpbird 247	. . 3 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> E. y e. { A , B } ( B ( +g ` M ) y ) =/= y )
39		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x ( +g ` M ) y ) = ( A ( +g ` M ) y ) )
40	39	neeq1d 2853	. . . . . 6 |- ( x = A -> ( ( x ( +g ` M ) y ) =/= y <-> ( A ( +g ` M ) y ) =/= y ) )
41	40	rexbidv 3052	. . . . 5 |- ( x = A -> ( E. y e. { A , B } ( x ( +g ` M ) y ) =/= y <-> E. y e. { A , B } ( A ( +g ` M ) y ) =/= y ) )
42		oveq1 6657	. . . . . . 7 |- ( x = B -> ( x ( +g ` M ) y ) = ( B ( +g ` M ) y ) )
43	42	neeq1d 2853	. . . . . 6 |- ( x = B -> ( ( x ( +g ` M ) y ) =/= y <-> ( B ( +g ` M ) y ) =/= y ) )
44	43	rexbidv 3052	. . . . 5 |- ( x = B -> ( E. y e. { A , B } ( x ( +g ` M ) y ) =/= y <-> E. y e. { A , B } ( B ( +g ` M ) y ) =/= y ) )
45	41, 44	ralprg 4234	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( A. x e. { A , B } E. y e. { A , B } ( x ( +g ` M ) y ) =/= y <-> ( E. y e. { A , B } ( A ( +g ` M ) y ) =/= y /\ E. y e. { A , B } ( B ( +g ` M ) y ) =/= y ) ) )
46	45	3adant3 1081	. . 3 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( A. x e. { A , B } E. y e. { A , B } ( x ( +g ` M ) y ) =/= y <-> ( E. y e. { A , B } ( A ( +g ` M ) y ) =/= y /\ E. y e. { A , B } ( B ( +g ` M ) y ) =/= y ) ) )
47	19, 38, 46	mpbir2and 957	. 2 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> A. x e. { A , B } E. y e. { A , B } ( x ( +g ` M ) y ) =/= y )
48	3, 1	eqtr2i 2645	. . 3 |- { A , B } = ( Base ` M )
49	48, 5	isnmnd 17298	. 2 |- ( A. x e. { A , B } E. y e. { A , B } ( x ( +g ` M ) y ) =/= y -> M e/ Mnd )
50	2, 47, 49	3syl 18	1 |- ( ( # ` S ) = 2 -> M e/ Mnd )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   ifcif 4086   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2c2 11070   #chash 13117   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294

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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mnd 17295

This theorem is referenced by:  sgrp2nmnd  17417  sgrpnmndex  17419