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Theorem sgrp2nmndlem3 18028
Description: Lemma 3 for sgrp2nmnd 18033. (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 2841 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 df-ne 3014 . . . . . 6 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
6 eqcom 2825 . . . . . . . . 9 (𝐴 = 𝑥𝑥 = 𝐴)
7 eqeq2 2830 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
87adantr 481 . . . . . . . . 9 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝑥𝐴 = 𝐵))
96, 8syl5rbbr 287 . . . . . . . 8 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝐵𝑥 = 𝐴))
109notbid 319 . . . . . . 7 ((𝑥 = 𝐵𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
1110biimpcd 250 . . . . . 6 𝐴 = 𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
125, 11sylbi 218 . . . . 5 (𝐴𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13123ad2ant3 1127 . . . 4 ((𝐶𝑆𝐵𝑆𝐴𝐵) → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
1413imp 407 . . 3 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
1514iffalsed 4474 . 2 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16 simp2 1129 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
17 simp1 1128 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐶𝑆)
184, 15, 16, 17, 16ovmpod 7291 1 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  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-ne 3014  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:  sgrp2rid2  18029  sgrp2nmndlem4  18031  sgrp2nmndlem5  18032
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