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Theorem mgmb1mgm1 18588
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 2726 . . . . . 6 (+𝑓𝑀) = (+𝑓𝑀)
41, 2, 3plusfeq 18581 . . . . 5 ( + Fn (𝐵 × 𝐵) → (+𝑓𝑀) = + )
51, 3mgmplusf 18583 . . . . . 6 (𝑀 ∈ Mgm → (+𝑓𝑀):(𝐵 × 𝐵)⟶𝐵)
6 feq1 6692 . . . . . 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 407 . . 3 ((𝑀 ∈ Mgm ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
1093adant2 1128 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶𝐵)
11 simp2 1134 . 2 ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → 𝑍𝐵)
12 intopsn 18587 . 2 (( + :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
1310, 11, 12syl2anc 583 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 1084   = wceq 1533  wcel 2098  {csn 4623  cop 4629   × cxp 5667   Fn wfn 6532  wf 6533  cfv 6537  Basecbs 17153  +gcplusg 17206  +𝑓cplusf 18570  Mgmcmgm 18571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-plusf 18572  df-mgm 18573
This theorem is referenced by:  srg1zr  20120
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