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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 19320 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Grp) |
4 | 2nn0 12493 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | hashbnd 14301 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) | |
6 | 4, 5 | mp3an2 1445 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐴 ∈ Fin) |
7 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
8 | 1, 7 | symghash 19297 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) = (!‘(♯‘𝐴))) |
10 | hashcl 14321 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘𝐴) ∈ ℕ0) |
12 | faccl 14248 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℕ) |
14 | 13 | nnred 12231 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ∈ ℝ) |
15 | 11, 11 | nn0expcld 14214 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℕ0) |
16 | 15 | nn0red 12537 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) ∈ ℝ) |
17 | 6re 12306 | . . . . 5 ⊢ 6 ∈ ℝ | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 6 ∈ ℝ) |
19 | facubnd 14265 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) | |
20 | 11, 19 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) ≤ ((♯‘𝐴)↑(♯‘𝐴))) |
21 | exple2lt6 47316 | . . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) | |
22 | 11, 21 | sylancom 587 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → ((♯‘𝐴)↑(♯‘𝐴)) < 6) |
23 | 14, 16, 18, 20, 22 | lelttrd 11376 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (!‘(♯‘𝐴)) < 6) |
24 | 9, 23 | eqbrtrd 5163 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → (♯‘(Base‘𝐺)) < 6) |
25 | 7 | lt6abl 19815 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (♯‘(Base‘𝐺)) < 6) → 𝐺 ∈ Abel) |
26 | 3, 24, 25 | syl2anc 583 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 ℝcr 11111 < clt 11252 ≤ cle 11253 ℕcn 12216 2c2 12271 6c6 12275 ℕ0cn0 12476 ↑cexp 14032 !cfa 14238 ♯chash 14295 Basecbs 17153 Grpcgrp 18863 SymGrpcsymg 19286 Abelcabl 19701 |
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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16779 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-tset 17225 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-efmnd 18794 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-eqg 19052 df-symg 19287 df-od 19448 df-gex 19449 df-cmn 19702 df-abl 19703 df-cyg 19798 |
This theorem is referenced by: (None) |
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