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Theorem mgm2nsgrplem2 17607
Description: Lemma 2 for mgm2nsgrp 17610. (Contributed by AV, 27-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
mgm2nsgrplem2 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
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 mgm2nsgrplem2
StepHypRef Expression
1 prid1g 4439 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2850 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4440 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2850 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2782 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 ifeq1 4234 . . . . . . 7 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
11 ifid 4269 . . . . . . 7 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1210, 11syl6eq 2810 . . . . . 6 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
1312a1d 25 . . . . 5 (𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
14 eqeq1 2764 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1514bicomd 213 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐵 = 𝐴𝑦 = 𝐴))
1615notbid 307 . . . . . . . . 9 (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
1716biimpac 504 . . . . . . . 8 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
1817intnand 1000 . . . . . . 7 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
1918iffalsed 4241 . . . . . 6 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2019ex 449 . . . . 5 𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2113, 20pm2.61i 176 . . . 4 (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2221ad2antll 767 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = (𝐴 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23 iftrue 4236 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
2423adantl 473 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25 simpl 474 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
26 simpr 479 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
279, 24, 25, 25, 26ovmpt2d 6953 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐵)
2827, 26eqeltrd 2839 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) ∈ 𝑆)
299, 22, 28, 26, 25ovmpt2d 6953 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐴) 𝐵) = 𝐴)
303, 5, 29syl2an 495 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  ifcif 4230  {cpr 4323  cfv 6049  (class class class)co 6813  cmpt2 6815  Basecbs 16059  +gcplusg 16143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818
This theorem is referenced by:  mgm2nsgrplem4  17609
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