Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pgrpgt2nabl Structured version   Visualization version   GIF version

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 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 𝐺 ∉ Abel)

Proof of Theorem pgrpgt2nabl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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)
41, 2, 3symgtrf 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)))
86, 7mpan 702 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) ↔ (2 + 1) ≤ (#‘𝐴)))
9 2p1e3 11028 . . . . . . . . . . . . . . . 16 (2 + 1) = 3
109a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
1110breq1d 4593 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (#‘𝐴) ↔ 3 ≤ (#‘𝐴)))
128, 11bitrd 267 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) ↔ 3 ≤ (#‘𝐴)))
1312biimpd 218 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℕ0 → (2 < (#‘𝐴) → 3 ≤ (#‘𝐴)))
1413adantld 482 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 → ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 3 ≤ (#‘𝐴)))
155, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 3 ≤ (#‘𝐴)))
16 3re 10971 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
1716rexri 9976 . . . . . . . . . . . . . . 15 3 ∈ ℝ*
18 pnfge 11840 . . . . . . . . . . . . . . 15 (3 ∈ ℝ* → 3 ≤ +∞)
1917, 18ax-mp 5 . . . . . . . . . . . . . 14 3 ≤ +∞
20 hashinf 12984 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞)
2119, 20syl5breqr 4621 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (#‘𝐴))
2221ex 449 . . . . . . . . . . . 12 (𝐴𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (#‘𝐴)))
2322adantr 480 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → (¬ 𝐴 ∈ Fin → 3 ≤ (#‘𝐴)))
2423com12 32 . . . . . . . . . 10 𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 3 ≤ (#‘𝐴)))
2515, 24pm2.61i 175 . . . . . . . . 9 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 3 ≤ (#‘𝐴))
26 eqid 2610 . . . . . . . . . . 11 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2726pmtr3ncom 17718 . . . . . . . . . 10 ((𝐴𝑉 ∧ 3 ≤ (#‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
28 rexcom 3080 . . . . . . . . . 10 (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) ↔ ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
2927, 28sylibr 223 . . . . . . . . 9 ((𝐴𝑉 ∧ 3 ≤ (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
3025, 29syldan 486 . . . . . . . 8 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
31 ssrexv 3630 . . . . . . . . 9 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
3231reximdv 2999 . . . . . . . 8 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
334, 30, 32mpsyl 66 . . . . . . 7 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥))
34 ssrexv 3630 . . . . . . 7 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
354, 33, 34mpsyl 66 . . . . . 6 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥))
36 eqid 2610 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
372, 3, 36symgov 17633 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
3837adantl 481 . . . . . . . 8 (((𝐴𝑉 ∧ 2 < (#‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
39 pm3.22 464 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
4039adantl 481 . . . . . . . . 9 (((𝐴𝑉 ∧ 2 < (#‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
412, 3, 36symgov 17633 . . . . . . . . 9 ((𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑥) = (𝑦𝑥))
4240, 41syl 17 . . . . . . . 8 (((𝐴𝑉 ∧ 2 < (#‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑥) = (𝑦𝑥))
4338, 42neeq12d 2843 . . . . . . 7 (((𝐴𝑉 ∧ 2 < (#‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ (𝑥𝑦) ≠ (𝑦𝑥)))
44432rexbidva 3038 . . . . . 6 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → (∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
4535, 44mpbird 246 . . . . 5 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
46 rexnal 2978 . . . . . 6 (∃𝑥 ∈ (Base‘𝐺) ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
47 rexnal 2978 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
48 df-ne 2782 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
4948bicomi 213 . . . . . . . . 9 (¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ (𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5049rexbii 3023 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5147, 50bitr3i 265 . . . . . . 7 (¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5251rexbii 3023 . . . . . 6 (∃𝑥 ∈ (Base‘𝐺) ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5346, 52bitr3i 265 . . . . 5 (¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5445, 53sylibr 223 . . . 4 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
5554intnand 953 . . 3 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
5655intnand 953 . 2 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
57 df-nel 2783 . . 3 (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58 isabl 18020 . . . 4 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
593, 36iscmn 18023 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
6059anbi2i 726 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6158, 60bitri 263 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6257, 61xchbinx 323 . 2 (𝐺 ∉ Abel ↔ ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6356, 62sylibr 223 1 ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 𝐺 ∉ Abel)
 Colors of variables: wff setvar class 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)
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