Users' Mathboxes 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 42665
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 2770 . . . . . . . 8 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2 pgrple2abl.g . . . . . . . 8 𝐺 = (SymGrp‘𝐴)
3 eqid 2770 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 2, 3symgtrf 18095 . . . . . . 7 ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5 hashcl 13348 . . . . . . . . . . 11 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6 2nn0 11510 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
7 nn0ltp1le 11636 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
86, 7mpan 662 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9 2p1e3 11352 . . . . . . . . . . . . . . . 16 (2 + 1) = 3
109a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
1110breq1d 4794 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
128, 11bitrd 268 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
1312biimpd 219 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
1413adantld 474 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
155, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
16 3re 11295 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
1716rexri 10298 . . . . . . . . . . . . . . 15 3 ∈ ℝ*
18 pnfge 12168 . . . . . . . . . . . . . . 15 (3 ∈ ℝ* → 3 ≤ +∞)
1917, 18ax-mp 5 . . . . . . . . . . . . . 14 3 ≤ +∞
20 hashinf 13325 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
2119, 20syl5breqr 4822 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
2221ex 397 . . . . . . . . . . . 12 (𝐴𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
2322adantr 466 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
2423com12 32 . . . . . . . . . 10 𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
2515, 24pm2.61i 176 . . . . . . . . 9 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴))
26 eqid 2770 . . . . . . . . . . 11 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2726pmtr3ncom 18101 . . . . . . . . . 10 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
28 rexcom 3246 . . . . . . . . . 10 (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) ↔ ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
2927, 28sylibr 224 . . . . . . . . 9 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
3025, 29syldan 571 . . . . . . . 8 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
31 ssrexv 3814 . . . . . . . . 9 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
3231reximdv 3163 . . . . . . . 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 68 . . . . . . 7 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥))
34 ssrexv 3814 . . . . . . 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 68 . . . . . 6 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥))
36 eqid 2770 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
372, 3, 36symgov 18016 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
3837adantl 467 . . . . . . . 8 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
39 pm3.22 449 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
4039adantl 467 . . . . . . . . 9 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
412, 3, 36symgov 18016 . . . . . . . . 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 3003 . . . . . . 7 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ (𝑥𝑦) ≠ (𝑦𝑥)))
44432rexbidva 3203 . . . . . 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 247 . . . . 5 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
46 rexnal 3142 . . . . . 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 3142 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
48 df-ne 2943 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
4948bicomi 214 . . . . . . . . 9 (¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ (𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5049rexbii 3188 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5147, 50bitr3i 266 . . . . . . 7 (¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5251rexbii 3188 . . . . . 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 266 . . . . 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 224 . . . 4 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
5554intnand 998 . . 3 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
5655intnand 998 . 2 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
57 df-nel 3046 . . 3 (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58 isabl 18403 . . . 4 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
593, 36iscmn 18406 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
6059anbi2i 601 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6158, 60bitri 264 . . 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 224 1 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wnel 3045  wral 3060  wrex 3061  wss 3721   class class class wbr 4784  ran crn 5250  ccom 5253  cfv 6031  (class class class)co 6792  Fincfn 8108  1c1 10138   + caddc 10140  +∞cpnf 10272  *cxr 10274   < clt 10275  cle 10276  2c2 11271  3c3 11272  0cn0 11493  chash 13320  Basecbs 16063  +gcplusg 16148  Mndcmnd 17501  Grpcgrp 17629  SymGrpcsymg 18003  pmTrspcpmtr 18067  CMndccmn 18399  Abelcabl 18400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-fz 12533  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-plusg 16161  df-tset 16167  df-symg 18004  df-pmtr 18068  df-cmn 18401  df-abl 18402
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator