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Mirrors > Home > MPE Home > Th. List > Mathboxes > pgrple2abl | Structured version Visualization version GIF version |
Description: Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.) |
Ref | Expression |
---|---|
pgrple2abl.g | ⊢ 𝐺 = (SymGrp‘𝐴) |
Ref | Expression |
---|---|
pgrple2abl | ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pgrple2abl.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | 1 | symggrp 18530 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Grp) |
4 | 2nn0 11917 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | hashbnd 13699 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) | |
6 | 4, 5 | mp3an2 1445 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) |
7 | eqid 2823 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
8 | 1, 7 | symghash 18508 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
10 | hashcl 13720 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘𝐴) ∈ ℕ0) |
12 | faccl 13646 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℕ) |
14 | 13 | nnred 11655 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℝ) |
15 | 11, 11 | nn0expcld 13610 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℕ0) |
16 | 15 | nn0red 11959 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℝ) |
17 | 6re 11730 | . . . . 5 ⊢ 6 ∈ ℝ | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 6 ∈ ℝ) |
19 | facubnd 13663 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) | |
20 | 11, 19 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) |
21 | exple2lt6 44419 | . . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) | |
22 | 11, 21 | sylancom 590 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) |
23 | 14, 16, 18, 20, 22 | lelttrd 10800 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) < 6) |
24 | 9, 23 | eqbrtrd 5090 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) < 6) |
25 | 7 | lt6abl 19017 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (♯‘(Base‘𝐺)) < 6) → 𝐺 ∈ Abel) |
26 | 3, 24, 25 | syl2anc 586 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℝcr 10538 < clt 10677 ≤ cle 10678 ℕcn 11640 2c2 11695 6c6 11699 ℕ0cn0 11900 ↑cexp 13432 !cfa 13636 ♯chash 13693 Basecbs 16485 Grpcgrp 18105 SymGrpcsymg 18497 Abelcabl 18909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-dvds 15610 df-gcd 15846 df-prm 16018 df-pc 16176 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-tset 16586 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-eqg 18280 df-symg 18498 df-od 18658 df-gex 18659 df-cmn 18910 df-abl 18911 df-cyg 18999 |
This theorem is referenced by: (None) |
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