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Theorem sgrp2nmndlem3 18090
Description: Lemma 3 for sgrp2nmnd 18095. (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))
sgrp2nmnd.p = (+g𝑀)
Assertion
Ref Expression
sgrp2nmndlem3 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥,𝐶,𝑦
Allowed substitution hints:   𝑀(𝑦)   (𝑥,𝑦)

Proof of Theorem sgrp2nmndlem3
StepHypRef Expression
1 sgrp2nmnd.p . . . 4 = (+g𝑀)
2 sgrp2nmnd.o . . . 4 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
31, 2eqtri 2847 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 df-ne 3015 . . . . . 6 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
6 eqeq2 2836 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
76adantr 484 . . . . . . . . 9 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝑥𝐴 = 𝐵))
8 eqcom 2831 . . . . . . . . 9 (𝐴 = 𝑥𝑥 = 𝐴)
97, 8bitr3di 289 . . . . . . . 8 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝐵𝑥 = 𝐴))
109notbid 321 . . . . . . 7 ((𝑥 = 𝐵𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
1110biimpcd 252 . . . . . 6 𝐴 = 𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
125, 11sylbi 220 . . . . 5 (𝐴𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13123ad2ant3 1132 . . . 4 ((𝐶𝑆𝐵𝑆𝐴𝐵) → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
1413imp 410 . . 3 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
1514iffalsed 4461 . 2 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16 simp2 1134 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
17 simp1 1133 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐶𝑆)
184, 15, 16, 17, 16ovmpod 7295 1 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  ifcif 4450  {cpr 4552  cfv 6343  (class class class)co 7149  cmpo 7151  Basecbs 16483  +gcplusg 16565
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154
This theorem is referenced by:  sgrp2rid2  18091  sgrp2nmndlem4  18093  sgrp2nmndlem5  18094
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