Theorem sgrp2nmndlem3 18986

Description: Lemma 3 for sgrp2nmnd 18991. (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
sgrp2nmndlem3	⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)

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 sgrp2nmndlem3

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 2792	. . 3 ⊢ ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
4	3	a1i 11	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5		df-ne 2965	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
6		eqeq2 2781	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
7	6	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
8		eqcom 2776	. . . . . . . . 9 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
9	7, 8	bitr3di 289	. . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝐵 ↔ 𝑥 = 𝐴))
10	9	notbid 321	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
11	10	biimpcd 252	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
12	5, 11	sylbi 220	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13	12	3ad2ant3 1151	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
14	13	imp 411	. . 3 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
15	14	iffalsed 4503	. 2 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16		simp2 1153	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
17		simp1 1152	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑆)
18	4, 15, 16, 17, 16	ovmpod 7563	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ifcif 4492  {cpr 4596  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17268  +_gcplusg 17309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  sgrp2rid2  18987  sgrp2nmndlem4  18989  sgrp2nmndlem5  18990