Theorem mgm2nsgrplem3 18933

Description: Lemma 3 for mgm2nsgrp 18935. (Contributed by AV, 28-Jan-2020.)

Hypotheses

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

Assertion

Ref	Expression
mgm2nsgrplem3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)

Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥, ⚬ ,𝑦

Allowed substitution hints:   𝑀(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mgm2nsgrplem3

Step	Hyp	Ref	Expression
1		prid1g 4760	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . 3 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	eleqtrrdi 2852	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4761	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	eleqtrrdi 2852	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆)
6		mgm2nsgrp.p	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
7		mgm2nsgrp.o	. . . . 5 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))
8	6, 7	eqtri 2765	. . . 4 ⊢ ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴))
9	8	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)))
10		simprl 771	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → 𝑥 = 𝐴)
11		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵)) → 𝑦 = (𝐴 ⚬ 𝐵))
12		ifeq1 4529	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴))
13		ifid 4566	. . . . . . . . . . 11 ⊢ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
14	12, 13	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
15	14	a1d 25	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
16		eqeq1 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐴 ↔ 𝐵 = 𝐴))
17	16	biimpcd 249	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 → 𝐵 = 𝐴))
18	17	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → (𝑦 = 𝐵 → 𝐵 = 𝐴))
19	18	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝐵 = 𝐴))
20	19	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝐵 = 𝐴))
21	20	con3d 152	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)))
22	21	impcom 407	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
23	22	iffalsed 4536	. . . . . . . . . 10 ⊢ ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
24	23	ex 412	. . . . . . . . 9 ⊢ (¬ 𝐵 = 𝐴 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
25	15, 24	pm2.61i 182	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
28		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
29	9, 26, 27, 28, 27	ovmpod 7585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) = 𝐴)
30	11, 29	sylan9eqr 2799	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → 𝑦 = 𝐴)
31	10, 30	jca 511	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
32	31	iftrued 4533	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
33	29, 27	eqeltrd 2841	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) ∈ 𝑆)
34	9, 32, 27, 33, 28	ovmpod 7585	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)
35	3, 5, 34	syl2an 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ifcif 4525  {cpr 4628  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17247  +_gcplusg 17297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  mgm2nsgrplem4  18934