Theorem mgmb1mgm1 12599

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 2165	. . . . . 6 |- ( +f ` M ) = ( +f ` M )
3	1, 2	mgmplusf 12597	. . . . 5 |- ( M e. Mgm -> ( +f ` M ) : ( B X. B ) --> B )
4	3	adantr 274	. . . 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 12595	. . . . 5 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( +f ` M ) = .+ )
7	6	feq1d 5324	. . . 4 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( ( +f ` M ) : ( B X. B ) --> B <-> .+ : ( B X. B ) --> B ) )
8	4, 7	mpbid 146	. . 3 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
9	8	3adant2 1006	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
10		simp2 988	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> Z e. B )
11		intopsn 12598	. 2 |- ( ( .+ : ( B X. B ) --> B /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
12	9, 10, 11	syl2anc 409	1 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {csn 3576   <.cop 3579   X. cxp 4602   Fn wfn 5183   -->wf 5184   ` cfv 5188   Basecbs 12394   +g cplusg 12457   +fcplusf 12584  Mgmcmgm 12585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-inn 8858  df-2 8916  df-ndx 12397  df-slot 12398  df-base 12400  df-plusg 12470  df-plusf 12586  df-mgm 12587

This theorem is referenced by: (None)