Theorem sgrp2nmndlem3 18960

Description: Lemma 3 for sgrp2nmnd 18965. (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 2768	. . 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 2947	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
6		eqeq2 2752	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
8		eqcom 2747	. . . . . . . . 9 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
9	7, 8	bitr3di 286	. . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝐵 ↔ 𝑥 = 𝐴))
10	9	notbid 318	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
11	10	biimpcd 249	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
12	5, 11	sylbi 217	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13	12	3ad2ant3 1135	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
14	13	imp 406	. . 3 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
15	14	iffalsed 4559	. 2 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16		simp2 1137	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
17		simp1 1136	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑆)
18	4, 15, 16, 17, 16	ovmpod 7602	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ifcif 4548  {cpr 4650  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Basecbs 17258  +_gcplusg 17311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  sgrp2rid2  18961  sgrp2nmndlem4  18963  sgrp2nmndlem5  18964