Theorem pgrpgt2nabl 49809

Description: Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)

Hypothesis

Ref	Expression
pgrple2abl.g	⊢ 𝐺 = (SymGrp‘𝐴)

Assertion

Ref	Expression
pgrpgt2nabl	⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)

Proof of Theorem pgrpgt2nabl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . . . 8 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2		pgrple2abl.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴)
3		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 2, 3	symgtrf 19734	. . . . . . 7 ⊢ ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5		hashcl 14541	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6		2nn0 12663	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
7		nn0ltp1le 12797	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
8	6, 7	mpan 703	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9		2p1e3 12524	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
10	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
11	10	breq1d 5118	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
12	8, 11	bitrd 282	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
13	12	biimpd 232	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
14	13	adantld 496	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
15	5, 14	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
16		3re 12463	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
17	16	rexri 11406	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ*
18		pnfge 13299	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ ℝ* → 3 ≤ +∞)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 3 ≤ +∞
20		hashinf 14520	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
21	19, 20	breqtrrid 5149	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
22	21	ex 418	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
23	22	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
24	23	com12 33	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
25	15, 24	pm2.61i 184	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴))
26		eqid 2763	. . . . . . . . . . 11 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
27	26	pmtr3ncom 19740	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
28		rexcom 3295	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) ↔ ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
29	27, 28	sylibr 237	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
30	25, 29	syldan 603	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
31		ssrexv 3983	. . . . . . . . 9 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
32	31	reximdv 3181	. . . . . . . 8 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
33	4, 30, 32	mpsyl 69	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
34		ssrexv 3983	. . . . . . 7 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
35	4, 33, 34	mpsyl 69	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
36		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
37	2, 3, 36	symgov 19649	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
38	37	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
39		pm3.22 465	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
40	39	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
41	2, 3, 36	symgov 19649	. . . . . . . . 9 ⊢ ((𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑥) = (𝑦 ∘ 𝑥))
42	40, 41	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑥) = (𝑦 ∘ 𝑥))
43	38, 42	neeq12d 3020	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ (𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
44	43	2rexbidva 3229	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → (∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
45	35, 44	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
46		rexnal 3118	. . . . . 6 ⊢ (∃𝑥 ∈ (Base‘𝐺) ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
47		rexnal 3118	. . . . . . . 8 ⊢ (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
48		df-ne 2960	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
49	48	bicomi 227	. . . . . . . . 9 ⊢ (¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
50	49	rexbii 3113	. . . . . . . 8 ⊢ (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
51	47, 50	bitr3i 280	. . . . . . 7 ⊢ (¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
52	51	rexbii 3113	. . . . . 6 ⊢ (∃𝑥 ∈ (Base‘𝐺) ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
53	46, 52	bitr3i 280	. . . . 5 ⊢ (¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
54	45, 53	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
55	54	intnand 494	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
56	55	intnand 494	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))))
57		df-nel 3066	. . 3 ⊢ (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58		isabl 20049	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
59	3, 36	iscmn 20054	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
60	59	anbi2i 635	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))))
61	58, 60	bitri 278	. . 3 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))))
62	57, 61	xchbinx 337	. 2 ⊢ (𝐺 ∉ Abel ↔ ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))))
63	56, 62	sylibr 237	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∉ wnel 3065  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108  ran crn 5659   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451  Fincfn 9011  1c1 11240   + caddc 11242  +∞cpnf 11379  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383  2c2 12437  3c3 12438  ℕ₀cn0 12646  ♯chash 14515  Basecbs 17431  +_gcplusg 17472  Mndcmnd 18973  Grpcgrp 19194  SymGrpcsymg 19634  pmTrspcpmtr 19706  CMndccmn 20045  Abelcabl 20046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-hash 14516  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-tset 17492  df-efmnd 19115  df-symg 19635  df-pmtr 19707  df-cmn 20047  df-abl 20048

This theorem is used by: (None)