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Theorem pgrpgt2nabl 42816
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 2765 . . . . . . . 8 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2 pgrple2abl.g . . . . . . . 8 𝐺 = (SymGrp‘𝐴)
3 eqid 2765 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 2, 3symgtrf 18152 . . . . . . 7 ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5 hashcl 13349 . . . . . . . . . . 11 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6 2nn0 11557 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
7 nn0ltp1le 11682 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
86, 7mpan 681 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9 2p1e3 11421 . . . . . . . . . . . . . . . 16 (2 + 1) = 3
109a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
1110breq1d 4819 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
128, 11bitrd 270 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
1312biimpd 220 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) → 3 ≤ (♯‘𝐴)))
1413adantld 484 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
155, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ Fin → ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴)))
16 3re 11352 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
1716rexri 10351 . . . . . . . . . . . . . . 15 3 ∈ ℝ*
18 pnfge 12164 . . . . . . . . . . . . . . 15 (3 ∈ ℝ* → 3 ≤ +∞)
1917, 18ax-mp 5 . . . . . . . . . . . . . 14 3 ≤ +∞
20 hashinf 13326 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
2119, 20syl5breqr 4847 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 3 ≤ (♯‘𝐴))
2221ex 401 . . . . . . . . . . . 12 (𝐴𝑉 → (¬ 𝐴 ∈ Fin → 3 ≤ (♯‘𝐴)))
2322adantr 472 . . . . . . . . . . 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 2765 . . . . . . . . . . 11 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2726pmtr3ncom 18158 . . . . . . . . . 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 225 . . . . . . . . 9 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
3025, 29syldan 585 . . . . . . . 8 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
31 ssrexv 3827 . . . . . . . . 9 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
3231reximdv 3162 . . . . . . . 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 3827 . . . . . . 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 2765 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
372, 3, 36symgov 18073 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
3837adantl 473 . . . . . . . 8 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥𝑦))
39 pm3.22 451 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
4039adantl 473 . . . . . . . . 9 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑦 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)))
412, 3, 36symgov 18073 . . . . . . . . 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 2998 . . . . . . 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 248 . . . . 5 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
46 rexnal 3141 . . . . . 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 3141 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
48 df-ne 2938 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
4948bicomi 215 . . . . . . . . 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 268 . . . . . . 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 268 . . . . 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 225 . . . 4 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
5554intnand 482 . . 3 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
5655intnand 482 . 2 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
57 df-nel 3041 . . 3 (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58 isabl 18463 . . . 4 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
593, 36iscmn 18466 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
6059anbi2i 616 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6158, 60bitri 266 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6257, 61xchbinx 325 . 2 (𝐺 ∉ Abel ↔ ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6356, 62sylibr 225 1 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wnel 3040  wral 3055  wrex 3056  wss 3732   class class class wbr 4809  ran crn 5278  ccom 5281  cfv 6068  (class class class)co 6842  Fincfn 8160  1c1 10190   + caddc 10192  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  2c2 11327  3c3 11328  0cn0 11538  chash 13321  Basecbs 16130  +gcplusg 16214  Mndcmnd 17560  Grpcgrp 17689  SymGrpcsymg 18060  pmTrspcpmtr 18124  CMndccmn 18459  Abelcabl 18460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-plusg 16227  df-tset 16233  df-symg 18061  df-pmtr 18125  df-cmn 18461  df-abl 18462
This theorem is referenced by: (None)
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