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Theorem pgrpgt2nabl 46754
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 2732 . . . . . . . 8 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2 pgrple2abl.g . . . . . . . 8 𝐺 = (SymGrp‘𝐴)
3 eqid 2732 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
41, 2, 3symgtrf 19303 . . . . . . 7 ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺)
5 hashcl 14300 . . . . . . . . . . 11 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
6 2nn0 12473 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
7 nn0ltp1le 12604 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
86, 7mpan 688 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ (2 + 1) ≤ (♯‘𝐴)))
9 2p1e3 12338 . . . . . . . . . . . . . . . 16 (2 + 1) = 3
109a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ0 → (2 + 1) = 3)
1110breq1d 5152 . . . . . . . . . . . . . 14 ((♯‘𝐴) ∈ ℕ0 → ((2 + 1) ≤ (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
128, 11bitrd 278 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 → (2 < (♯‘𝐴) ↔ 3 ≤ (♯‘𝐴)))
1312biimpd 228 . . . . . . . . . . . 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 12276 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
1716rexri 11256 . . . . . . . . . . . . . . 15 3 ∈ ℝ*
18 pnfge 13094 . . . . . . . . . . . . . . 15 (3 ∈ ℝ* → 3 ≤ +∞)
1917, 18ax-mp 5 . . . . . . . . . . . . . 14 3 ≤ +∞
20 hashinf 14279 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
2119, 20breqtrrid 5180 . . . . . . . . . . . . 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 182 . . . . . . . . 9 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 3 ≤ (♯‘𝐴))
26 eqid 2732 . . . . . . . . . . 11 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2726pmtr3ncom 19309 . . . . . . . . . 10 ((𝐴𝑉 ∧ 3 ≤ (♯‘𝐴)) → ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
28 rexcom 3287 . . . . . . . . . 10 (∃𝑥 ∈ ran (pmTrsp‘𝐴)∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) ↔ ∃𝑦 ∈ ran (pmTrsp‘𝐴)∃𝑥 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥))
2927, 28sylibr 233 . . . . . . . . 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 4048 . . . . . . . . 9 (ran (pmTrsp‘𝐴) ⊆ (Base‘𝐺) → (∃𝑦 ∈ ran (pmTrsp‘𝐴)(𝑥𝑦) ≠ (𝑦𝑥) → ∃𝑦 ∈ (Base‘𝐺)(𝑥𝑦) ≠ (𝑦𝑥)))
3231reximdv 3170 . . . . . . . 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 4048 . . . . . . 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 2732 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
372, 3, 36symgov 19217 . . . . . . . . 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 19217 . . . . . . . . 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 3002 . . . . . . 7 (((𝐴𝑉 ∧ 2 < (♯‘𝐴)) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ (𝑥𝑦) ≠ (𝑦𝑥)))
44432rexbidva 3217 . . . . . 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 256 . . . . 5 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
46 rexnal 3100 . . . . . 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 3100 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
48 df-ne 2941 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥) ↔ ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
4948bicomi 223 . . . . . . . . 9 (¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ (𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5049rexbii 3094 . . . . . . . 8 (∃𝑦 ∈ (Base‘𝐺) ¬ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5147, 50bitr3i 276 . . . . . . 7 (¬ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ≠ (𝑦(+g𝐺)𝑥))
5251rexbii 3094 . . . . . 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 276 . . . . 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 233 . . . 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 3047 . . 3 (𝐺 ∉ Abel ↔ ¬ 𝐺 ∈ Abel)
58 isabl 19618 . . . 4 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
593, 36iscmn 19623 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
6059anbi2i 623 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6158, 60bitri 274 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6257, 61xchbinx 333 . 2 (𝐺 ∉ Abel ↔ ¬ (𝐺 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
6356, 62sylibr 233 1 ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wnel 3046  wral 3061  wrex 3070  wss 3945   class class class wbr 5142  ran crn 5671  ccom 5674  cfv 6533  (class class class)co 7394  Fincfn 8924  1c1 11095   + caddc 11097  +∞cpnf 11229  *cxr 11231   < clt 11232  cle 11233  2c2 12251  3c3 12252  0cn0 12456  chash 14274  Basecbs 17128  +gcplusg 17181  Mndcmnd 18604  Grpcgrp 18796  SymGrpcsymg 19200  pmTrspcpmtr 19275  CMndccmn 19614  Abelcabl 19615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-oadd 8454  df-er 8688  df-map 8807  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-dju 9880  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-xnn0 12529  df-z 12543  df-uz 12807  df-fz 13469  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-tset 17200  df-efmnd 18727  df-symg 19201  df-pmtr 19276  df-cmn 19616  df-abl 19617
This theorem is referenced by: (None)
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