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Theorem sgrp2nmndlem2 18735
Description: Lemma 2 for sgrp2nmnd 18741. (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 2765 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐴𝑆𝐶𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 iftrue 4493 . . 3 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
65ad2antrl 727 . 2 (((𝐴𝑆𝐶𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7 simpl 484 . 2 ((𝐴𝑆𝐶𝑆) → 𝐴𝑆)
8 simpr 486 . 2 ((𝐴𝑆𝐶𝑆) → 𝐶𝑆)
94, 6, 7, 8, 7ovmpod 7508 1 ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ifcif 4487  {cpr 4589  cfv 6497  (class class class)co 7358  cmpo 7360  Basecbs 17084  +gcplusg 17134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  sgrp2rid2  18737  sgrp2nmndlem4  18739  sgrp2nmndlem5  18740
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