Theorem mgmb1mgm1 17254

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 2622	. . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
4	1, 2, 3	plusfeq 17249	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝑀) = + )
5	1, 3	mgmplusf 17251	. . . . . 6 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵)
6		feq1 6026	. . . . . 6 ⊢ ((+𝑓‘𝑀) = + → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵))
7	5, 6	syl5ib 234	. . . . 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 446	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10	9	3adant2 1080	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
11		simp2 1062	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵)
12		intopsn 17253	. 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13	10, 11, 12	syl2anc 693	1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {csn 4177  ⟨cop 4183   × cxp 5112   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  Basecbs 15857  +_gcplusg 15941  +_𝑓cplusf 17239  Mgmcmgm 17240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-plusf 17241  df-mgm 17242

This theorem is referenced by:  srg1zr  18529