Theorem mgmb1mgm1 13011

Description: The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)

Hypotheses

Ref	Expression
mgmb1mgm1.b	|- B = ( Base ` M )
mgmb1mgm1.p	|- .+ = ( +g ` M )

Assertion

Ref	Expression
mgmb1mgm1	|- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )

Proof of Theorem mgmb1mgm1

Step	Hyp	Ref	Expression
1		mgmb1mgm1.b	. . . . . 6 |- B = ( Base ` M )
2		eqid 2196	. . . . . 6 |- ( +f ` M ) = ( +f ` M )
3	1, 2	mgmplusf 13009	. . . . 5 |- ( M e. Mgm -> ( +f ` M ) : ( B X. B ) --> B )
4	3	adantr 276	. . . 4 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( +f ` M ) : ( B X. B ) --> B )
5		mgmb1mgm1.p	. . . . . 6 |- .+ = ( +g ` M )
6	1, 5, 2	plusfeqg 13007	. . . . 5 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( +f ` M ) = .+ )
7	6	feq1d 5394	. . . 4 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( ( +f ` M ) : ( B X. B ) --> B <-> .+ : ( B X. B ) --> B ) )
8	4, 7	mpbid 147	. . 3 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
9	8	3adant2 1018	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
10		simp2 1000	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> Z e. B )
11		intopsn 13010	. 2 |- ( ( .+ : ( B X. B ) --> B /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
12	9, 10, 11	syl2anc 411	1 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   {csn 3622   <.cop 3625   X. cxp 4661   Fn wfn 5253   -->wf 5254   ` cfv 5258   Basecbs 12678   +g cplusg 12755   +fcplusf 12996  Mgmcmgm 12997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-plusf 12998  df-mgm 12999

This theorem is referenced by:  srg1zr  13543