Theorem mgmb1mgm1 13450

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 2231	. . . . . 6 |- ( +f ` M ) = ( +f ` M )
3	1, 2	mgmplusf 13448	. . . . 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 13446	. . . . 5 |- ( ( M e. Mgm /\ .+ Fn ( B X. B ) ) -> ( +f ` M ) = .+ )
7	6	feq1d 5469	. . . 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 1042	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) --> B )
10		simp2 1024	. 2 |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> Z e. B )
11		intopsn 13449	. 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 1004   = wceq 1397   e. wcel 2202   {csn 3669   <.cop 3672   X. cxp 4723   Fn wfn 5321   -->wf 5322   ` cfv 5326   Basecbs 13081   +g cplusg 13159   +fcplusf 13435  Mgmcmgm 13436

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-plusf 13437  df-mgm 13438

This theorem is referenced by:  srg1zr  13999