Theorem mgm2nsgrplem2 18853

Description: Lemma 2 for mgm2nsgrp 18856. (Contributed by AV, 27-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
mgm2nsgrplem2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)

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 mgm2nsgrplem2

Step	Hyp	Ref	Expression
1		prid1g 4727	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . 3 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	eleqtrrdi 2840	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4728	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	eleqtrrdi 2840	. 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 2753	. . . 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		ifeq1 4495	. . . . . . 7 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴))
11		ifid 4532	. . . . . . 7 ⊢ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
12	10, 11	eqtrdi 2781	. . . . . 6 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
13	12	a1d 25	. . . . 5 ⊢ (𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
14		eqeq1 2734	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐴 ↔ 𝐵 = 𝐴))
15	14	bicomd 223	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝐴 ↔ 𝑦 = 𝐴))
16	15	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
17	16	biimpac 478	. . . . . . . 8 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
18	17	intnand 488	. . . . . . 7 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
19	18	iffalsed 4502	. . . . . 6 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
20	19	ex 412	. . . . 5 ⊢ (¬ 𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
21	13, 20	pm2.61i 182	. . . 4 ⊢ (𝑦 = 𝐵 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
22	21	ad2antll 729	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = (𝐴 ⚬ 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23		iftrue 4497	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
24	23	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
26		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
27	9, 24, 25, 25, 26	ovmpod 7544	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐵)
28	27, 26	eqeltrd 2829	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) ∈ 𝑆)
29	9, 22, 28, 26, 25	ovmpod 7544	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)
30	3, 5, 29	syl2an 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ifcif 4491  {cpr 4594  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17186  +_gcplusg 17227

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  mgm2nsgrplem4  18855