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Theorem sgrp2nmndlem2 18027
Description: Lemma 2 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
sgrp2nmndlem2 ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)
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 sgrp2nmndlem2
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 ((𝐴𝑆𝐶𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 iftrue 4469 . . 3 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
65ad2antrl 724 . 2 (((𝐴𝑆𝐶𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7 simpl 483 . 2 ((𝐴𝑆𝐶𝑆) → 𝐴𝑆)
8 simpr 485 . 2 ((𝐴𝑆𝐶𝑆) → 𝐶𝑆)
94, 6, 7, 8, 7ovmpod 7291 1 ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  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:  sgrp2rid2  18029  sgrp2nmndlem4  18031  sgrp2nmndlem5  18032
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