Theorem mgmb1mgm1 13315

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 2207	. . . . . 6 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
3	1, 2	mgmplusf 13313	. . . . 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 13311	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝑀) = + )
7	6	feq1d 5432	. . . 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 1019	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10		simp2 1001	. 2 ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵)
11		intopsn 13314	. 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 981   = wceq 1373   ∈ wcel 2178  {csn 3643  ⟨cop 3646   × cxp 4691   Fn wfn 5285  ⟶wf 5286  ‘cfv 5290  Basecbs 12947  +_gcplusg 13024  +_𝑓cplusf 13300  Mgmcmgm 13301

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-plusf 13302  df-mgm 13303

This theorem is referenced by:  srg1zr  13864