Theorem sgrp2nmndlem2 18981

Description: Lemma 2 for sgrp2nmnd 18987. (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 2786	. . 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 4493	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
6	5	ad2antrl 740	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7		simpl 487	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐴 ∈ 𝑆)
8		simpr 489	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐶 ∈ 𝑆)
9	4, 6, 7, 8, 7	ovmpod 7562	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ⚬ 𝐶) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ifcif 4487  {cpr 4591  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  +_gcplusg 17305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  sgrp2rid2  18983  sgrp2nmndlem4  18985  sgrp2nmndlem5  18986