Theorem mgmb1mgm1 12679

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 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Proof of Theorem mgmb1mgm1

Step	Hyp	Ref	Expression
1		mgmb1mgm1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2177	. . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
3	1, 2	mgmplusf 12677	. . . . 5 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵)
4	3	adantr 276	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵)
5		mgmb1mgm1.p	. . . . . 6 ⊢ + = (+g‘𝑀)
6	1, 5, 2	plusfeqg 12675	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝑀) = + )
7	6	feq1d 5348	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵))
8	4, 7	mpbid 147	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
9	8	3adant2 1016	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10		simp2 998	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵)
11		intopsn 12678	. 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
12	9, 10, 11	syl2anc 411	1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {csn 3591  ⟨cop 3594   × cxp 4621   Fn wfn 5207  ⟶wf 5208  ‘cfv 5212  Basecbs 12445  +_gcplusg 12518  +_𝑓cplusf 12664  Mgmcmgm 12665

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-plusf 12666  df-mgm 12667

This theorem is referenced by:  srg1zr  12996