Theorem sgrp2nmndlem2 19025

Description: Lemma 2 for sgrp2nmnd 19031. (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 2788	. . 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 4495	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
6	5	ad2antrl 741	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7		simpl 488	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐴 ∈ 𝑆)
8		simpr 490	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐶 ∈ 𝑆)
9	4, 6, 7, 8, 7	ovmpod 7571	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ⚬ 𝐶) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ifcif 4489  {cpr 4593  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Basecbs 17293  +_gcplusg 17334

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424

This theorem is used by:  sgrp2rid2  19027  sgrp2nmndlem4  19029  sgrp2nmndlem5  19030