Theorem pgrpgt2nabl 46432

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 2736	. . . . . . . 8 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2		pgrple2abl.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴)
3		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 2, 3	symgtrf 19251	. . . . . . 7 ⊢ ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5		hashcl 14256	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6		2nn0 12430	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
7		nn0ltp1le 12561	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
8	6, 7	mpan 688	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9		2p1e3 12295	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
10	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
11	10	breq1d 5115	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
12	8, 11	bitrd 278	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
13	12	biimpd 228	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
14	13	adantld 491	. . . . . . . . . . 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 12233	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
17	16	rexri 11213	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ*
18		pnfge 13051	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ ℝ* → 3 ≤ +∞)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 3 ≤ +∞
20		hashinf 14235	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
21	19, 20	breqtrrid 5143	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
22	21	ex 413	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
23	22	adantr 481	. . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
27	26	pmtr3ncom 19257	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
28		rexcom 3273	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
30	25, 29	syldan 591	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
31		ssrexv 4011	. . . . . . . . 9 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
32	31	reximdv 3167	. . . . . . . 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 4011	. . . . . . 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 2736	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
37	2, 3, 36	symgov 19165	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
38	37	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥 ∘ 𝑦))
39		pm3.22 460	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
40	39	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
41	2, 3, 36	symgov 19165	. . . . . . . . 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 3005	. . . . . . 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 3211	. . . . . 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 256	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
46		rexnal 3103	. . . . . 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 3103	. . . . . . . 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 2944	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
49	48	bicomi 223	. . . . . . . . 9 ⊢ (¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
50	49	rexbii 3097	. . . . . . . 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 276	. . . . . . 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 3097	. . . . . 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 276	. . . . 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 233	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
55	54	intnand 489	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
56	55	intnand 489	. 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 3050	. . 3 ⊢ (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58		isabl 19566	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
59	3, 36	iscmn 19571	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
60	59	anbi2i 623	. . . 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 274	. . 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 333	. 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 233	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3073   ⊆ wss 3910   class class class wbr 5105  ran crn 5634   ∘ ccom 5637  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052   + caddc 11054  +∞cpnf 11186  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190  2c2 12208  3c3 12209  ℕ₀cn0 12413  ♯chash 14230  Basecbs 17083  +_gcplusg 17133  Mndcmnd 18556  Grpcgrp 18748  SymGrpcsymg 19148  pmTrspcpmtr 19223  CMndccmn 19562  Abelcabl 19563

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-tset 17152  df-efmnd 18679  df-symg 19149  df-pmtr 19224  df-cmn 19564  df-abl 19565

This theorem is referenced by: (None)