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Theorem mgmb1mgm1 18511
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 mgmb1mgm1.p . . . . . 6 + = (+g𝑀)
3 eqid 2737 . . . . . 6 (+𝑓𝑀) = (+𝑓𝑀)
41, 2, 3plusfeq 18506 . . . . 5 ( + Fn (𝐵 × 𝐵) → (+𝑓𝑀) = + )
51, 3mgmplusf 18508 . . . . . 6 (𝑀 ∈ Mgm → (+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵)
6 feq1 6650 . . . . . 6 ((+𝑓𝑀) = + → ((+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵+ :(𝐵 × 𝐵)⟶𝐵))
75, 6imbitrid 243 . . . . 5 ((+𝑓𝑀) = + → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
84, 7syl 17 . . . 4 ( + Fn (𝐵 × 𝐵) → (𝑀 ∈ Mgm → + :(𝐵 × 𝐵)⟶𝐵))
98impcom 409 . . 3 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
1093adant2 1132 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
11 simp2 1138 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → 𝑍𝐵)
12 intopsn 18510 . 2 (( + :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
1310, 11, 12syl2anc 585 1 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wcel 2107  {csn 4587  cop 4593   × cxp 5632   Fn wfn 6492  wf 6493  cfv 6497  Basecbs 17084  +gcplusg 17134  +𝑓cplusf 18495  Mgmcmgm 18496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-plusf 18497  df-mgm 18498
This theorem is referenced by:  srg1zr  19947
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