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Theorem mgmb1mgm1 12816
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 + = (+g𝑀)
Assertion
Ref Expression
mgmb1mgm1 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Proof of Theorem mgmb1mgm1
StepHypRef Expression
1 mgmb1mgm1.b . . . . . 6 𝐵 = (Base‘𝑀)
2 eqid 2189 . . . . . 6 (+𝑓𝑀) = (+𝑓𝑀)
31, 2mgmplusf 12814 . . . . 5 (𝑀 ∈ Mgm → (+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵)
43adantr 276 . . . 4 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵)
5 mgmb1mgm1.p . . . . . 6 + = (+g𝑀)
61, 5, 2plusfeqg 12812 . . . . 5 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓𝑀) = + )
76feq1d 5367 . . . 4 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → ((+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵+ :(𝐵 × 𝐵)⟶𝐵))
84, 7mpbid 147 . . 3 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
983adant2 1018 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10 simp2 1000 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → 𝑍𝐵)
11 intopsn 12815 . 2 (( + :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
129, 10, 11syl2anc 411 1 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2160  {csn 3607  cop 3610   × cxp 4639   Fn wfn 5226  wf 5227  cfv 5231  Basecbs 12486  +gcplusg 12561  +𝑓cplusf 12801  Mgmcmgm 12802
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-plusg 12574  df-plusf 12803  df-mgm 12804
This theorem is referenced by:  srg1zr  13308
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