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Theorem mgm2nsgrplem3 18097
 Description: Lemma 3 for mgm2nsgrp 18099. (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 4659 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2901 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4660 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2901 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2821 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 simprl 770 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑥 = 𝐴)
11 simpr 488 . . . . . 6 ((𝑥 = 𝐴𝑦 = (𝐴 𝐵)) → 𝑦 = (𝐴 𝐵))
12 ifeq1 4432 . . . . . . . . . . 11 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
13 ifid 4467 . . . . . . . . . . 11 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1412, 13eqtrdi 2849 . . . . . . . . . 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 2802 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1716biimpcd 252 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐵 = 𝐴))
1817adantl 485 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑦 = 𝐵𝐵 = 𝐴))
1918com12 32 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2019adantl 485 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2120con3d 155 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴𝑦 = 𝐴)))
2221impcom 411 . . . . . . . . . . 11 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
2322iffalsed 4439 . . . . . . . . . 10 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2423ex 416 . . . . . . . . 9 𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2515, 24pm2.61i 185 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2625adantl 485 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27 simpl 486 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
28 simpr 488 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
299, 26, 27, 28, 27ovmpod 7292 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
3011, 29sylan9eqr 2855 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑦 = 𝐴)
3110, 30jca 515 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → (𝑥 = 𝐴𝑦 = 𝐴))
3231iftrued 4436 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
3329, 27eqeltrd 2890 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) ∈ 𝑆)
349, 32, 27, 33, 28ovmpod 7292 . 2 ((𝐴𝑆𝐵𝑆) → (𝐴 (𝐴 𝐵)) = 𝐵)
353, 5, 34syl2an 598 1 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ifcif 4428  {cpr 4530  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  Basecbs 16495  +gcplusg 16577 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150 This theorem is referenced by:  mgm2nsgrplem4  18098
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