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Theorem sgrp2nmndlem1 17233
 Description: Lemma 1 for sgrp2nmnd 17240: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17077). (Contributed by AV, 29-Jan-2020.)
Hypotheses
Ref Expression
mgm2nsgrp.s 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b (Base‘𝑀) = 𝑆
sgrp2nmnd.o (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
Assertion
Ref Expression
sgrp2nmndlem1 ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀
Allowed substitution hints:   𝑀(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sgrp2nmndlem1
StepHypRef Expression
1 prid1g 4239 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2699 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4240 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2699 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.b . . . 4 (Base‘𝑀) = 𝑆
76eqcomi 2619 . . 3 𝑆 = (Base‘𝑀)
8 sgrp2nmnd.o . . 3 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
9 ne0i 3880 . . . 4 (𝐴𝑆𝑆 ≠ ∅)
109adantr 480 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑆 ≠ ∅)
11 simpll 786 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥𝑆𝑦𝑆)) → 𝐴𝑆)
12 simplr 788 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥𝑆𝑦𝑆)) → 𝐵𝑆)
137, 8, 10, 11, 12opifismgm 17081 . 2 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
143, 5, 13syl2an 493 1 ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ifcif 4036  {cpr 4127  ‘cfv 5804   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mgm 17065 This theorem is referenced by:  sgrp2nmndlem4  17238
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