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Mirrors > Home > MPE Home > Th. List > odsubdvds | Structured version Visualization version GIF version |
Description: The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
odsubdvds.1 | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
odsubdvds | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) ∥ (♯‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 18570 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
3 | 2 | 3ad2ant1 1135 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (𝐺 ↾s 𝑆) ∈ Grp) |
4 | 1 | subgbas 18571 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
5 | 4 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
6 | simp2 1139 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → 𝑆 ∈ Fin) | |
7 | 5, 6 | eqeltrrd 2840 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (Base‘(𝐺 ↾s 𝑆)) ∈ Fin) |
8 | simp3 1140 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆) | |
9 | 8, 5 | eleqtrd 2841 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ (Base‘(𝐺 ↾s 𝑆))) |
10 | eqid 2738 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
11 | eqid 2738 | . . . 4 ⊢ (od‘(𝐺 ↾s 𝑆)) = (od‘(𝐺 ↾s 𝑆)) | |
12 | 10, 11 | oddvds2 18981 | . . 3 ⊢ (((𝐺 ↾s 𝑆) ∈ Grp ∧ (Base‘(𝐺 ↾s 𝑆)) ∈ Fin ∧ 𝐴 ∈ (Base‘(𝐺 ↾s 𝑆))) → ((od‘(𝐺 ↾s 𝑆))‘𝐴) ∥ (♯‘(Base‘(𝐺 ↾s 𝑆)))) |
13 | 3, 7, 9, 12 | syl3anc 1373 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → ((od‘(𝐺 ↾s 𝑆))‘𝐴) ∥ (♯‘(Base‘(𝐺 ↾s 𝑆)))) |
14 | odsubdvds.1 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
15 | 1, 14, 11 | subgod 18983 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) = ((od‘(𝐺 ↾s 𝑆))‘𝐴)) |
16 | 15 | 3adant2 1133 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) = ((od‘(𝐺 ↾s 𝑆))‘𝐴)) |
17 | 5 | fveq2d 6739 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (♯‘𝑆) = (♯‘(Base‘(𝐺 ↾s 𝑆)))) |
18 | 13, 16, 17 | 3brtr4d 5099 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) ∥ (♯‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 class class class wbr 5067 ‘cfv 6397 (class class class)co 7231 Fincfn 8646 ♯chash 13920 ∥ cdvds 15839 Basecbs 16784 ↾s cress 16808 Grpcgrp 18389 SubGrpcsubg 18561 odcod 18940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-disj 5033 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-oadd 8226 df-omul 8227 df-er 8411 df-ec 8413 df-qs 8417 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-sup 9082 df-inf 9083 df-oi 9150 df-card 9579 df-acn 9582 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-n0 12115 df-z 12201 df-uz 12463 df-rp 12611 df-fz 13120 df-fzo 13263 df-fl 13391 df-mod 13467 df-seq 13599 df-exp 13660 df-hash 13921 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-clim 15073 df-sum 15274 df-dvds 15840 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-0g 16970 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-submnd 18243 df-grp 18392 df-minusg 18393 df-sbg 18394 df-mulg 18513 df-subg 18564 df-eqg 18566 df-od 18944 |
This theorem is referenced by: odcau 19017 ablfac1eu 19484 idomsubgmo 40754 |
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