Theorem sgrp2nmndlem2 17234

Description: Lemma 2 for sgrp2nmnd 17240. (Contributed by AV, 29-Jan-2020.)

Hypotheses

Ref	Expression
mgm2nsgrp.s	⊢ 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b	⊢ (Base‘𝑀) = 𝑆
sgrp2nmnd.o	⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
sgrp2nmnd.p	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
sgrp2nmndlem2	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (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

Step	Hyp	Ref	Expression
1		sgrp2nmnd.p	. . . 4 ⊢ ⚬ = (+g‘𝑀)
2		sgrp2nmnd.o	. . . 4 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
3	1, 2	eqtri 2632	. . 3 ⊢ ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
4	3	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5		iftrue 4042	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
6	5	ad2antrl 760	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7		simpl 472	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐴 ∈ 𝑆)
8		simpr 476	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐶 ∈ 𝑆)
9	4, 6, 7, 8, 7	ovmpt2d 6686	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ⚬ 𝐶) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036  {cpr 4127  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +_gcplusg 15768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  sgrp2rid2  17236  sgrp2nmndlem4  17238  sgrp2nmndlem5  17239