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Mirrors > Home > MPE Home > Th. List > odhash2 | Structured version Visualization version GIF version |
Description: If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
odhash.x | ⊢ 𝑋 = (Base‘𝐺) |
odhash.o | ⊢ 𝑂 = (od‘𝐺) |
odhash.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
odhash2 | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(𝐾‘{𝐴})) = (𝑂‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odhash.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2818 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | odhash.o | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
4 | odhash.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
5 | 1, 2, 3, 4 | odf1o2 18627 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 ∈ (0..^(𝑂‘𝐴)) ↦ (𝑥(.g‘𝐺)𝐴)):(0..^(𝑂‘𝐴))–1-1-onto→(𝐾‘{𝐴})) |
6 | ovex 7178 | . . . 4 ⊢ (0..^(𝑂‘𝐴)) ∈ V | |
7 | 6 | f1oen 8518 | . . 3 ⊢ ((𝑥 ∈ (0..^(𝑂‘𝐴)) ↦ (𝑥(.g‘𝐺)𝐴)):(0..^(𝑂‘𝐴))–1-1-onto→(𝐾‘{𝐴}) → (0..^(𝑂‘𝐴)) ≈ (𝐾‘{𝐴})) |
8 | hasheni 13696 | . . 3 ⊢ ((0..^(𝑂‘𝐴)) ≈ (𝐾‘{𝐴}) → (♯‘(0..^(𝑂‘𝐴))) = (♯‘(𝐾‘{𝐴}))) | |
9 | 5, 7, 8 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(0..^(𝑂‘𝐴))) = (♯‘(𝐾‘{𝐴}))) |
10 | 1, 3 | odcl 18593 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ0) |
11 | 10 | 3ad2ant2 1126 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ0) |
12 | hashfzo0 13779 | . . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ0 → (♯‘(0..^(𝑂‘𝐴))) = (𝑂‘𝐴)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(0..^(𝑂‘𝐴))) = (𝑂‘𝐴)) |
14 | 9, 13 | eqtr3d 2855 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(𝐾‘{𝐴})) = (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {csn 4557 class class class wbr 5057 ↦ cmpt 5137 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ≈ cen 8494 0cc0 10525 ℕcn 11626 ℕ0cn0 11885 ..^cfzo 13021 ♯chash 13678 Basecbs 16471 mrClscmrc 16842 Grpcgrp 18041 .gcmg 18162 SubGrpcsubg 18211 odcod 18581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-od 18585 |
This theorem is referenced by: odhash3 18630 proot1mul 39677 |
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