Theorem mgm2nsgrplem3 17407

Description: Lemma 3 for mgm2nsgrp 17409. (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 4295	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . 3 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	syl6eleqr 2712	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4296	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	syl6eleqr 2712	. 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 2644	. . . 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 794	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → 𝑥 = 𝐴)
11		simpr 477	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵)) → 𝑦 = (𝐴 ⚬ 𝐵))
12		ifeq1 4090	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐴 → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴))
13		ifid 4125	. . . . . . . . . . 11 ⊢ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
14	12, 13	syl6eq 2672	. . . . . . . . . 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 2626	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐴 ↔ 𝐵 = 𝐴))
17	16	biimpcd 239	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 → 𝐵 = 𝐴))
18	17	adantl 482	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝐵 = 𝐴))
21	20	con3d 148	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)))
22	21	impcom 446	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
23	22	iffalsed 4097	. . . . . . . . . 10 ⊢ ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
24	23	ex 450	. . . . . . . . 9 ⊢ (¬ 𝐵 = 𝐴 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
25	15, 24	pm2.61i 176	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
26	25	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27		simpl 473	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
28		simpr 477	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
29	9, 26, 27, 28, 27	ovmpt2d 6788	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) = 𝐴)
30	11, 29	sylan9eqr 2678	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → 𝑦 = 𝐴)
31	10, 30	jca 554	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
32	31	iftrued 4094	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 = 𝐴 ∧ 𝑦 = (𝐴 ⚬ 𝐵))) → if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
33	29, 27	eqeltrd 2701	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) ∈ 𝑆)
34	9, 32, 27, 33, 28	ovmpt2d 6788	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)
35	3, 5, 34	syl2an 494	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ifcif 4086  {cpr 4179  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  mgm2nsgrplem4  17408