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Theorem sgrp2nmndlem1 17350
Description: Lemma 1 for sgrp2nmnd 17357: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17194). (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 4272 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2709 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4273 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2709 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.b . . . 4 (Base‘𝑀) = 𝑆
76eqcomi 2630 . . 3 𝑆 = (Base‘𝑀)
8 sgrp2nmnd.o . . 3 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
9 ne0i 3903 . . . 4 (𝐴𝑆𝑆 ≠ ∅)
109adantr 481 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑆 ≠ ∅)
11 simpll 789 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥𝑆𝑦𝑆)) → 𝐴𝑆)
12 simplr 791 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥𝑆𝑦𝑆)) → 𝐵𝑆)
137, 8, 10, 11, 12opifismgm 17198 . 2 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
143, 5, 13syl2an 494 1 ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  c0 3897  ifcif 4064  {cpr 4157  cfv 5857  cmpt2 6617  Basecbs 15800  +gcplusg 15881  Mgmcmgm 17180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-mgm 17182
This theorem is referenced by:  sgrp2nmndlem4  17355
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