Theorem mgmb1mgm1 12622

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 2170	. . . . . 6 |- ( +f ` M ) = ( +f ` M )
3	1, 2	mgmplusf 12620	. . . . 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 12618	. . . . 5 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( +f ` M ) = .+ )
7	6	feq1d 5334	. . . 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 1011	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
10		simp2 993	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> Z e. B )
11		intopsn 12621	. 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 973   = wceq 1348   e. wcel 2141   {csn 3583   <.cop 3586   X. cxp 4609   Fn wfn 5193   -->wf 5194   ` cfv 5198   Basecbs 12416   +g cplusg 12480   +fcplusf 12607  Mgmcmgm 12608

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-plusf 12609  df-mgm 12610

This theorem is referenced by: (None)