Theorem mgm2nsgrplem1 17405

Description: Lemma 1 for mgm2nsgrp 17409: M is a magma, even if A = B ( M is the trivial magma in this case, see mgmb1mgm1 17254). (Contributed by AV, 27-Jan-2020.)

Hypotheses

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

Assertion

Ref	Expression
mgm2nsgrplem1	|- ( ( A e. V /\ B e. W ) -> M e. Mgm )

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

Allowed substitution hints:   M( y)   V( x, y)   W( x, y)

Proof of Theorem mgm2nsgrplem1

Step	Hyp	Ref	Expression
1		prid1g 4295	. . 3 |- ( A e. V -> A e. { A , B } )
2		mgm2nsgrp.s	. . 3 |- S = { A , B }
3	1, 2	syl6eleqr 2712	. 2 |- ( A e. V -> A e. S )
4		prid2g 4296	. . 3 |- ( B e. W -> B e. { A , B } )
5	4, 2	syl6eleqr 2712	. 2 |- ( B e. W -> B e. S )
6		mgm2nsgrp.b	. . . 4 |- ( Base ` M ) = S
7	6	eqcomi 2631	. . 3 |- S = ( Base ` M )
8		mgm2nsgrp.o	. . 3 |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )
9		ne0i 3921	. . . 4 |- ( A e. S -> S =/= (/) )
10	9	adantr 481	. . 3 |- ( ( A e. S /\ B e. S ) -> S =/= (/) )
11		simplr 792	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( x e. S /\ y e. S ) ) -> B e. S )
12		simpll 790	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( x e. S /\ y e. S ) ) -> A e. S )
13	7, 8, 10, 11, 12	opifismgm 17258	. 2 |- ( ( A e. S /\ B e. S ) -> M e. Mgm )
14	3, 5, 13	syl2an 494	1 |- ( ( A e. V /\ B e. W ) -> M e. Mgm )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   ifcif 4086   {cpr 4179   ` cfv 5888   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240

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

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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-mgm 17242

This theorem is referenced by:  mgm2nsgrp  17409  mgmnsgrpex  17418