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Theorem mgm2nsgrplem3 18023
Description: Lemma 3 for mgm2nsgrp 18025. (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 4688 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2921 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4689 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2921 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2841 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 simprl 767 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑥 = 𝐴)
11 simpr 485 . . . . . 6 ((𝑥 = 𝐴𝑦 = (𝐴 𝐵)) → 𝑦 = (𝐴 𝐵))
12 ifeq1 4467 . . . . . . . . . . 11 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
13 ifid 4502 . . . . . . . . . . 11 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1412, 13syl6eq 2869 . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1716biimpcd 250 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐵 = 𝐴))
1817adantl 482 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑦 = 𝐵𝐵 = 𝐴))
1918com12 32 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2019adantl 482 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2120con3d 155 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴𝑦 = 𝐴)))
2221impcom 408 . . . . . . . . . . 11 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
2322iffalsed 4474 . . . . . . . . . 10 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2423ex 413 . . . . . . . . 9 𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2515, 24pm2.61i 183 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2625adantl 482 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27 simpl 483 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
28 simpr 485 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
299, 26, 27, 28, 27ovmpod 7291 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
3011, 29sylan9eqr 2875 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑦 = 𝐴)
3110, 30jca 512 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → (𝑥 = 𝐴𝑦 = 𝐴))
3231iftrued 4471 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
3329, 27eqeltrd 2910 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) ∈ 𝑆)
349, 32, 27, 33, 28ovmpod 7291 . 2 ((𝐴𝑆𝐵𝑆) → (𝐴 (𝐴 𝐵)) = 𝐵)
353, 5, 34syl2an 595 1 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  ifcif 4463  {cpr 4559  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  +gcplusg 16553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  mgm2nsgrplem4  18024
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