Theorem pgrpgt2nabl 48091

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 2740	. . . . . . . 8 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2		pgrple2abl.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴)
3		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 2, 3	symgtrf 19511	. . . . . . 7 ⊢ ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5		hashcl 14405	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6		2nn0 12570	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
7		nn0ltp1le 12701	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
8	6, 7	mpan 689	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9		2p1e3 12435	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
10	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
11	10	breq1d 5176	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
12	8, 11	bitrd 279	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
13	12	biimpd 229	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
14	13	adantld 490	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
16		3re 12373	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
17	16	rexri 11348	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ*
18		pnfge 13193	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ ℝ* → 3 ≤ +∞)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 3 ≤ +∞
20		hashinf 14384	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
21	19, 20	breqtrrid 5204	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
22	21	ex 412	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
24	23	com12 32	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
25	15, 24	pm2.61i 182	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴))
26		eqid 2740	. . . . . . . . . . 11 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
27	26	pmtr3ncom 19517	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
28		rexcom 3296	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
30	25, 29	syldan 590	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
31		ssrexv 4078	. . . . . . . . 9 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
32	31	reximdv 3176	. . . . . . . 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 68	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
34		ssrexv 4078	. . . . . . 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 68	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
36		eqid 2740	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
37	2, 3, 36	symgov 19425	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
38	37	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
39		pm3.22 459	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
40	39	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
41	2, 3, 36	symgov 19425	. . . . . . . . 9 ⊢ ((𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑥) = (𝑦 ∘ 𝑥))
42	40, 41	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑥) = (𝑦 ∘ 𝑥))
43	38, 42	neeq12d 3008	. . . . . . 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 3226	. . . . . 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 257	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
46		rexnal 3106	. . . . . 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 3106	. . . . . . . 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 2947	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
49	48	bicomi 224	. . . . . . . . 9 ⊢ (¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
50	49	rexbii 3100	. . . . . . . 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 277	. . . . . . 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 3100	. . . . . 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 277	. . . . 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 234	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
55	54	intnand 488	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
56	55	intnand 488	. 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 3053	. . 3 ⊢ (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58		isabl 19826	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
59	3, 36	iscmn 19831	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
60	59	anbi2i 622	. . . 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 275	. . 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 334	. 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 234	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   class class class wbr 5166  ran crn 5701   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  1c1 11185   + caddc 11187  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  2c2 12348  3c3 12349  ℕ₀cn0 12553  ♯chash 14379  Basecbs 17258  +_gcplusg 17311  Mndcmnd 18772  Grpcgrp 18973  SymGrpcsymg 19410  pmTrspcpmtr 19483  CMndccmn 19822  Abelcabl 19823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-tset 17330  df-efmnd 18904  df-symg 19411  df-pmtr 19484  df-cmn 19824  df-abl 19825

This theorem is referenced by: (None)