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Mathbox for Alexander van der Vekens |
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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 19135 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | 2 | adantr 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Grp) |
4 | 2nn0 12388 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | hashbnd 14190 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) | |
6 | 4, 5 | mp3an2 1449 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) |
7 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
8 | 1, 7 | symghash 19112 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
10 | hashcl 14210 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘𝐴) ∈ ℕ0) |
12 | faccl 14137 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℕ) |
14 | 13 | nnred 12126 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℝ) |
15 | 11, 11 | nn0expcld 14103 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℕ0) |
16 | 15 | nn0red 12432 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℝ) |
17 | 6re 12201 | . . . . 5 ⊢ 6 ∈ ℝ | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 6 ∈ ℝ) |
19 | facubnd 14154 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) | |
20 | 11, 19 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) |
21 | exple2lt6 46335 | . . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) | |
22 | 11, 21 | sylancom 588 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) |
23 | 14, 16, 18, 20, 22 | lelttrd 11271 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) < 6) |
24 | 9, 23 | eqbrtrd 5125 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) < 6) |
25 | 7 | lt6abl 19625 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (♯‘(Base‘𝐺)) < 6) → 𝐺 ∈ Abel) |
26 | 3, 24, 25 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 Fincfn 8841 ℝcr 11008 < clt 11147 ≤ cle 11148 ℕcn 12111 2c2 12166 6c6 12170 ℕ0cn0 12371 ↑cexp 13921 !cfa 14127 ♯chash 14184 Basecbs 17037 Grpcgrp 18702 SymGrpcsymg 19101 Abelcabl 19516 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-disj 5069 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-omul 8409 df-er 8606 df-ec 8608 df-qs 8612 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-dju 9795 df-card 9833 df-acn 9836 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-xnn0 12444 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-fz 13379 df-fzo 13522 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-fac 14128 df-bc 14157 df-hash 14185 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-clim 15324 df-sum 15525 df-dvds 16091 df-gcd 16329 df-prm 16502 df-pc 16663 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-tset 17106 df-0g 17277 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-efmnd 18633 df-grp 18705 df-minusg 18706 df-sbg 18707 df-mulg 18826 df-subg 18878 df-eqg 18880 df-symg 19102 df-od 19263 df-gex 19264 df-cmn 19517 df-abl 19518 df-cyg 19608 |
This theorem is referenced by: (None) |
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