Theorem mgmb1mgm1 17077

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		mgmb1mgm1.p	. . . . . 6 ⊢ + = (+g‘𝑀)
3		eqid 2610	. . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
4	1, 2, 3	plusfeq 17072	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝑀) = + )
5	1, 3	mgmplusf 17074	. . . . . 6 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵)
6		feq1 5939	. . . . . 6 ⊢ ((+𝑓‘𝑀) = + → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵))
7	5, 6	syl5ib 233	. . . . 5 ⊢ ((+𝑓‘𝑀) = + → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
8	4, 7	syl 17	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
9	8	impcom 445	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10	9	3adant2 1073	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
11		simp2 1055	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵)
12		intopsn 17076	. 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13	10, 11, 12	syl2anc 691	1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131   × cxp 5036   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  +_gcplusg 15768  +_𝑓cplusf 17062  Mgmcmgm 17063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-plusf 17064  df-mgm 17065

This theorem is referenced by:  srg1zr  18352