Theorem mgm2nsgrplem2 18876

Description: Lemma 2 for mgm2nsgrp 18879. (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 4767	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . 3 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	eleqtrrdi 2839	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4768	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	eleqtrrdi 2839	. 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 2755	. . . 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 4534	. . . . . . 7 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴))
11		ifid 4570	. . . . . . 7 ⊢ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
12	10, 11	eqtrdi 2783	. . . . . 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 2731	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐴 ↔ 𝐵 = 𝐴))
15	14	bicomd 222	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝐴 ↔ 𝑦 = 𝐴))
16	15	notbid 317	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
17	16	biimpac 477	. . . . . . . 8 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
18	17	intnand 487	. . . . . . 7 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
19	18	iffalsed 4541	. . . . . 6 ⊢ ((¬ 𝐵 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
20	19	ex 411	. . . . 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 727	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = (𝐴 ⚬ 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23		iftrue 4536	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
24	23	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25		simpl 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
26		simpr 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
27	9, 24, 25, 25, 26	ovmpod 7577	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐵)
28	27, 26	eqeltrd 2828	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) ∈ 𝑆)
29	9, 22, 28, 26, 25	ovmpod 7577	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)
30	3, 5, 29	syl2an 594	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ifcif 4530  {cpr 4632  ‘cfv 6551  (class class class)co 7424   ∈ cmpo 7426  Basecbs 17185  +_gcplusg 17238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429

This theorem is referenced by:  mgm2nsgrplem4  18878