Theorem mgmb1mgm1 18716

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 2770	. . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
4	1, 2, 3	plusfeq 18709	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝑀) = + )
5	1, 3	mgmplusf 18711	. . . . . 6 ⊢ (𝑀 ∈ Mgm → (+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵)
6		feq1 6687	. . . . . 6 ⊢ ((+𝑓‘𝑀) = + → ((+𝑓‘𝑀):(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶𝐵))
7	5, 6	imbitrid 247	. . . . 5 ⊢ ((+𝑓‘𝑀) = + → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
8	4, 7	syl 18	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
9	8	impcom 412	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10	9	3adant2 1147	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
11		simp2 1153	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵)
12		intopsn 18715	. 2 ⊢ (( + :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13	10, 11, 12	syl2anc 595	1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  {csn 4594  ⟨cop 4600   × cxp 5663   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  Basecbs 17272  +_gcplusg 17313  +_𝑓cplusf 18698  Mgmcmgm 18699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-plusf 18700  df-mgm 18701

This theorem is referenced by:  rng1zrlem  20262