Theorem sgrp2rid2 18565

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 4696	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2		mgm2nsgrp.s	. . . 4 ⊢ 𝑆 = {𝐴, 𝐵}
3	1, 2	eleqtrrdi 2850	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆)
4		prid2g 4697	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
5	4, 2	eleqtrrdi 2850	. . 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 18563	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴)
11	6, 10	syldan 591	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴)
12		oveq1 7282	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐴))
13		id 22	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
14	12, 13	eqeq12d 2754	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐴) = 𝐵))
15	11, 14	syl5ib 243	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵))
16		simprl 768	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆)
17		simprr 770	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆)
18		neqne 2951	. . . . . . . 8 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
19	18	adantr 481	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ≠ 𝐵)
20	2, 7, 8, 9	sgrp2nmndlem3 18564	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐴) = 𝐵)
21	16, 17, 19, 20	syl3anc 1370	. . . . . 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 18563	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) = 𝐴)
25	13, 13	oveq12d 7293	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐵))
26	25, 13	eqeq12d 2754	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐵) = 𝐵))
27	11, 26	syl5ib 243	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵))
28	2, 7, 8, 9	sgrp2nmndlem3 18564	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐵) = 𝐵)
29	17, 17, 19, 28	syl3anc 1370	. . . . . . 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 3346	. . . . 5 ⊢ (∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 ⚬ 𝑥) = 𝑦)
36		oveq1 7282	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ⚬ 𝑥) = (𝐴 ⚬ 𝑥))
37		id 22	. . . . . . 7 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
38	36, 37	eqeq12d 2754	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐴 ⚬ 𝑥) = 𝐴))
39		oveq1 7282	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ⚬ 𝑥) = (𝐵 ⚬ 𝑥))
40		id 22	. . . . . . 7 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
41	39, 40	eqeq12d 2754	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐵 ⚬ 𝑥) = 𝐵))
42	38, 41	ralprg 4630	. . . . 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 3112	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵)))
45	2	raleqi 3346	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵))
46		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐴))
47	46	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐴) = 𝐴))
48		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐴))
49	48	eqeq1d 2740	. . . . . 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 7283	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐵))
52	51	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐵) = 𝐴))
53		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐵))
54	53	eqeq1d 2740	. . . . . 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 4630	. . . 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 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ifcif 4459  {cpr 4563  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  +_gcplusg 16962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  sgrp2rid2ex  18566