Theorem sgrp2rid2 18809

Description: A small semigroup (with two elements) with two right identities which are different if 𝐴 ≠ 𝐵. (Contributed by AV, 10-Feb-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
sgrp2rid2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦)

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

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

Proof of Theorem sgrp2rid2

Step	Hyp	Ref	Expression
1		prid1g 4764	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . . 4 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	eleqtrrdi 2844	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4765	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	eleqtrrdi 2844	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆)
6		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
7		mgm2nsgrp.b	. . . . . 6 ⊢ (Base‘𝑀) = 𝑆
8		sgrp2nmnd.o	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
9		sgrp2nmnd.p	. . . . . 6 ⊢ ⚬ = (+g‘𝑀)
10	2, 7, 8, 9	sgrp2nmndlem2 18807	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴)
11	6, 10	syldan 591	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴)
12		oveq1 7418	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐴))
13		id 22	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
14	12, 13	eqeq12d 2748	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐴) = 𝐵))
15	11, 14	imbitrid 243	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵))
16		simprl 769	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆)
17		simprr 771	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆)
18		neqne 2948	. . . . . . . 8 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
19	18	adantr 481	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ≠ 𝐵)
20	2, 7, 8, 9	sgrp2nmndlem3 18808	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐴) = 𝐵)
21	16, 17, 19, 20	syl3anc 1371	. . . . . 6 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐵 ⚬ 𝐴) = 𝐵)
22	21	ex 413	. . . . 5 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵))
23	15, 22	pm2.61i 182	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵)
24	2, 7, 8, 9	sgrp2nmndlem2 18807	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) = 𝐴)
25	13, 13	oveq12d 7429	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐵))
26	25, 13	eqeq12d 2748	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐵) = 𝐵))
27	11, 26	imbitrid 243	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵))
28	2, 7, 8, 9	sgrp2nmndlem3 18808	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐵) = 𝐵)
29	17, 17, 19, 28	syl3anc 1371	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐵 ⚬ 𝐵) = 𝐵)
30	29	ex 413	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵))
31	27, 30	pm2.61i 182	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵)
32	24, 31	jca 512	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))
33	11, 23, 32	jca31 515	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))
34	3, 5, 33	syl2an 596	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))
35	2	raleqi 3323	. . . . 5 ⊢ (∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 ⚬ 𝑥) = 𝑦)
36		oveq1 7418	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ⚬ 𝑥) = (𝐴 ⚬ 𝑥))
37		id 22	. . . . . . 7 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
38	36, 37	eqeq12d 2748	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐴 ⚬ 𝑥) = 𝐴))
39		oveq1 7418	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ⚬ 𝑥) = (𝐵 ⚬ 𝑥))
40		id 22	. . . . . . 7 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
41	39, 40	eqeq12d 2748	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐵 ⚬ 𝑥) = 𝐵))
42	38, 41	ralprg 4698	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑦 ∈ {𝐴, 𝐵} (𝑦 ⚬ 𝑥) = 𝑦 ↔ ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵)))
43	35, 42	bitrid 282	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵)))
44	43	ralbidv 3177	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵)))
45	2	raleqi 3323	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵))
46		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐴))
47	46	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐴) = 𝐴))
48		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐴))
49	48	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐵 ⚬ 𝑥) = 𝐵 ↔ (𝐵 ⚬ 𝐴) = 𝐵))
50	47, 49	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵)))
51		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐵))
52	51	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐵) = 𝐴))
53		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐵))
54	53	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐵 ⚬ 𝑥) = 𝐵 ↔ (𝐵 ⚬ 𝐵) = 𝐵))
55	52, 54	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))
56	50, 55	ralprg 4698	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))))
57	45, 56	bitrid 282	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))))
58	44, 57	bitrd 278	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))))
59	34, 58	mpbird 256	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ifcif 4528  {cpr 4630  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17146  +_gcplusg 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  sgrp2rid2ex  18810