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Theorem mgm2nsgrplem3 17230
 Description: Lemma 3 for mgm2nsgrp 17232. (Contributed by AV, 28-Jan-2020.)
Hypotheses
Ref Expression
mgm2nsgrp.s 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b (Base‘𝑀) = 𝑆
mgm2nsgrp.o (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
mgm2nsgrp.p = (+g𝑀)
Assertion
Ref Expression
mgm2nsgrplem3 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥, ,𝑦
Allowed substitution hints:   𝑀(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mgm2nsgrplem3
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.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2632 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 simprl 790 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑥 = 𝐴)
11 simpr 476 . . . . . 6 ((𝑥 = 𝐴𝑦 = (𝐴 𝐵)) → 𝑦 = (𝐴 𝐵))
12 ifeq1 4040 . . . . . . . . . . 11 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
13 ifid 4075 . . . . . . . . . . 11 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1412, 13syl6eq 2660 . . . . . . . . . 10 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
1514a1d 25 . . . . . . . . 9 (𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
16 eqeq1 2614 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1716biimpcd 238 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐵 = 𝐴))
1817adantl 481 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑦 = 𝐵𝐵 = 𝐴))
1918com12 32 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2019adantl 481 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2120con3d 147 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴𝑦 = 𝐴)))
2221impcom 445 . . . . . . . . . . 11 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
2322iffalsed 4047 . . . . . . . . . 10 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2423ex 449 . . . . . . . . 9 𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2515, 24pm2.61i 175 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2625adantl 481 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27 simpl 472 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
28 simpr 476 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
299, 26, 27, 28, 27ovmpt2d 6686 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
3011, 29sylan9eqr 2666 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑦 = 𝐴)
3110, 30jca 553 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → (𝑥 = 𝐴𝑦 = 𝐴))
3231iftrued 4044 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
3329, 27eqeltrd 2688 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) ∈ 𝑆)
349, 32, 27, 33, 28ovmpt2d 6686 . 2 ((𝐴𝑆𝐵𝑆) → (𝐴 (𝐴 𝐵)) = 𝐵)
353, 5, 34syl2an 493 1 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036  {cpr 4127  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  mgm2nsgrplem4  17231
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