Theorem pgrpgt2nabl 41941

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 2610	. . . . . . . 8 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2		pgrple2abl.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴)
3		eqid 2610	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 2, 3	symgtrf 17712	. . . . . . 7 ⊢ ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5		hashcl 13009	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
6		2nn0 11186	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
7		nn0ltp1le 11312	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (2 < (#‘𝐴) ↔ (2 + 1) ≤ (#‘𝐴)))
8	6, 7	mpan 702	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) ↔ (2 + 1) ≤ (#‘𝐴)))
9		2p1e3 11028	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
10	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
11	10	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (#‘𝐴) ↔ 3 ≤ (#‘𝐴)))
12	8, 11	bitrd 267	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) ↔ 3 ≤ (#‘𝐴)))
13	12	biimpd 218	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) → 3 ≤ (#‘𝐴)))
14	13	adantld 482	. . . . . . . . . . 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 10971	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
17	16	rexri 9976	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ*
18		pnfge 11840	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ ℝ* → 3 ≤ +∞)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 3 ≤ +∞
20		hashinf 12984	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞)
21	19, 20	syl5breqr 4621	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (#‘𝐴))
22	21	ex 449	. . . . . . . . . . . 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 175	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → 3 ≤ (#‘𝐴))
26		eqid 2610	. . . . . . . . . . 11 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
27	26	pmtr3ncom 17718	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (#‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
28		rexcom 3080	. . . . . . . . . 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 223	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 3 ≤ (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
30	25, 29	syldan 486	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
31		ssrexv 3630	. . . . . . . . 9 ⊢ (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥)))
32	31	reximdv 2999	. . . . . . . 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 66	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
34		ssrexv 3630	. . . . . . 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 66	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥 ∘ 𝑦) ≠ (𝑦 ∘ 𝑥))
36		eqid 2610	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
37	2, 3, 36	symgov 17633	. . . . . . . . 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		pm3.22 464	. . . . . . . . . 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 17633	. . . . . . . . 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 2843	. . . . . . 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 3038	. . . . . 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 246	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
46		rexnal 2978	. . . . . 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 2978	. . . . . . . 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 2782	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥) ↔ ¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
49	48	bicomi 213	. . . . . . . . 9 ⊢ (¬ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (𝑥(+g‘𝐺)𝑦) ≠ (𝑦(+g‘𝐺)𝑥))
50	49	rexbii 3023	. . . . . . . 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 265	. . . . . . 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 3023	. . . . . 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 265	. . . . 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 223	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
55	54	intnand 953	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
56	55	intnand 953	. 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 2783	. . 3 ⊢ (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58		isabl 18020	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
59	3, 36	iscmn 18023	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
60	59	anbi2i 726	. . . 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 263	. . 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 323	. 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 223	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (#‘𝐴)) → 𝐺 ∉ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583  ran crn 5039   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  2c2 10947  3c3 10948  ℕ₀cn0 11169  #chash 12979  Basecbs 15695  +_gcplusg 15768  Mndcmnd 17117  Grpcgrp 17245  SymGrpcsymg 17620  pmTrspcpmtr 17684  CMndccmn 18016  Abelcabl 18017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-symg 17621  df-pmtr 17685  df-cmn 18018  df-abl 18019

This theorem is referenced by: (None)