Theorem pgrpgt2nabl 48554

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 2734	. . . . . . . 8 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2		pgrple2abl.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴)
3		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 2, 3	symgtrf 19396	. . . . . . 7 ⊢ ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5		hashcl 14277	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6		2nn0 12416	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
7		nn0ltp1le 12548	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
8	6, 7	mpan 690	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9		2p1e3 12280	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
10	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
11	10	breq1d 5106	. . . . . . . . . . . . . 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 12223	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
17	16	rexri 11188	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ*
18		pnfge 13042	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ ℝ* → 3 ≤ +∞)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 3 ≤ +∞
20		hashinf 14256	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
21	19, 20	breqtrrid 5134	. . . . . . . . . . . . 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 2734	. . . . . . . . . . 11 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
27	26	pmtr3ncom 19402	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
28		rexcom 3263	. . . . . . . . . 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 591	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
31		ssrexv 4001	. . . . . . . . 9 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
32	31	reximdv 3149	. . . . . . . 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 4001	. . . . . . 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 2734	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
37	2, 3, 36	symgov 19311	. . . . . . . . 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 19311	. . . . . . . . 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 2991	. . . . . . 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 3197	. . . . . 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 3086	. . . . . 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 3086	. . . . . . . 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 2931	. . . . . . . . . 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 3081	. . . . . . . 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 3081	. . . . . 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 3035	. . 3 ⊢ (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58		isabl 19711	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
59	3, 36	iscmn 19716	. . . . 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 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 1541   ∈ wcel 2113   ≠ wne 2930   ∉ wnel 3034  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899   class class class wbr 5096  ran crn 5623   ∘ ccom 5626  ‘cfv 6490  (class class class)co 7356  Fincfn 8881  1c1 11025   + caddc 11027  +∞cpnf 11161  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165  2c2 12198  3c3 12199  ℕ₀cn0 12399  ♯chash 14251  Basecbs 17134  +_gcplusg 17175  Mndcmnd 18657  Grpcgrp 18861  SymGrpcsymg 19296  pmTrspcpmtr 19368  CMndccmn 19707  Abelcabl 19708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-xnn0 12473  df-z 12487  df-uz 12750  df-fz 13422  df-hash 14252  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-tset 17194  df-efmnd 18792  df-symg 19297  df-pmtr 19369  df-cmn 19709  df-abl 19710

This theorem is referenced by: (None)