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Theorem mgmb1mgm1 12805
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 2187 . . . . . 6 (+𝑓𝑀) = (+𝑓𝑀)
31, 2mgmplusf 12803 . . . . 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 12801 . . . . 5 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → (+𝑓𝑀) = + )
76feq1d 5364 . . . 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 1017 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
10 simp2 999 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → 𝑍𝐵)
11 intopsn 12804 . 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 979   = wceq 1363  wcel 2158  {csn 3604  cop 3607   × cxp 4636   Fn wfn 5223  wf 5224  cfv 5228  Basecbs 12475  +gcplusg 12550  +𝑓cplusf 12790  Mgmcmgm 12791
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-plusf 12792  df-mgm 12793
This theorem is referenced by:  srg1zr  13224
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