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Theorem pgrpgt2nabl 44316
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 2826 . . . . . . . 8 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2 pgrple2abl.g . . . . . . . 8 𝐺 = (SymGrp‘𝐴)
3 eqid 2826 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 2, 3symgtrf 18533 . . . . . . 7 ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5 hashcl 13712 . . . . . . . . . . 11 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6 2nn0 11908 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
7 nn0ltp1le 12034 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
86, 7mpan 686 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9 2p1e3 11773 . . . . . . . . . . . . . . . 16 (2 + 1) = 3
109a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
1110breq1d 5073 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
128, 11bitrd 280 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
1312biimpd 230 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
1413adantld 491 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
155, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
16 3re 11711 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
1716rexri 10693 . . . . . . . . . . . . . . 15 3 ∈ ℝ*
18 pnfge 12520 . . . . . . . . . . . . . . 15 (3 ∈ ℝ* → 3 ≤ +∞)
1917, 18ax-mp 5 . . . . . . . . . . . . . 14 3 ≤ +∞
20 hashinf 13690 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
2119, 20breqtrrid 5101 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
2221ex 413 . . . . . . . . . . . 12 (𝐴𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
2322adantr 481 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
2423com12 32 . . . . . . . . . 10 𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
2515, 24pm2.61i 183 . . . . . . . . 9 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴))
26 eqid 2826 . . . . . . . . . . 11 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2726pmtr3ncom 18539 . . . . . . . . . 10 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
28 rexcom 3360 . . . . . . . . . 10 (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) ↔ ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
2927, 28sylibr 235 . . . . . . . . 9 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
3025, 29syldan 591 . . . . . . . 8 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
31 ssrexv 4038 . . . . . . . . 9 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
3231reximdv 3278 . . . . . . . 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 4038 . . . . . . 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 2826 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
372, 3, 36symgov 18453 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
3837adantl 482 . . . . . . . 8 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
39 pm3.22 460 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
4039adantl 482 . . . . . . . . 9 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
412, 3, 36symgov 18453 . . . . . . . . 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 3082 . . . . . . 7 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ (𝑥𝑦) ≠ (𝑦𝑥)))
44432rexbidva 3304 . . . . . 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 258 . . . . 5 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
46 rexnal 3243 . . . . . 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 3243 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
48 df-ne 3022 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
4948bicomi 225 . . . . . . . . 9 (¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ (𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5049rexbii 3252 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5147, 50bitr3i 278 . . . . . . 7 (¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5251rexbii 3252 . . . . . 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 278 . . . . 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 235 . . . 4 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
5554intnand 489 . . 3 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
5655intnand 489 . 2 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
57 df-nel 3129 . . 3 (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58 isabl 18846 . . . 4 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
593, 36iscmn 18850 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
6059anbi2i 622 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6158, 60bitri 276 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6257, 61xchbinx 335 . 2 (𝐺 ∉ Abel ↔ ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6356, 62sylibr 235 1 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wnel 3128  wral 3143  wrex 3144  wss 3940   class class class wbr 5063  ran crn 5555  ccom 5558  cfv 6354  (class class class)co 7150  Fincfn 8503  1c1 10532   + caddc 10534  +∞cpnf 10666  *cxr 10668   < clt 10669  cle 10670  2c2 11686  3c3 11687  0cn0 11891  chash 13685  Basecbs 16478  +gcplusg 16560  Mndcmnd 17906  Grpcgrp 18048  SymGrpcsymg 18440  pmTrspcpmtr 18505  CMndccmn 18842  Abelcabl 18843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12888  df-hash 13686  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-tset 16579  df-symg 18441  df-pmtr 18506  df-cmn 18844  df-abl 18845
This theorem is referenced by: (None)
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