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Mirrors > Home > MPE Home > Th. List > oddvds2 | Structured version Visualization version GIF version |
Description: The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
odcl2.1 | ⊢ 𝑋 = (Base‘𝐺) |
odcl2.2 | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
oddvds2 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ (♯‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odcl2.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
2 | odcl2.2 | . . . . 5 ⊢ 𝑂 = (od‘𝐺) | |
3 | eqid 2732 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | eqid 2732 | . . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) | |
5 | 1, 2, 3, 4 | dfod2 19426 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = if(ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ Fin, (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))), 0)) |
6 | 5 | 3adant2 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = if(ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ Fin, (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))), 0)) |
7 | simp2 1137 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ Fin) | |
8 | 1, 3, 4 | cycsubgcl 19077 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)))) |
9 | 8 | 3adant2 1131 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)))) |
10 | 9 | simpld 495 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺)) |
11 | 1 | subgss 19001 | . . . . . 6 ⊢ (ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ⊆ 𝑋) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ⊆ 𝑋) |
13 | 7, 12 | ssfid 9263 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ Fin) |
14 | 13 | iftrued 4535 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → if(ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ Fin, (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))), 0) = (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)))) |
15 | 6, 14 | eqtrd 2772 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)))) |
16 | 1 | lagsubg 19066 | . . 3 ⊢ ((ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))) ∥ (♯‘𝑋)) |
17 | 10, 7, 16 | syl2anc 584 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))) ∥ (♯‘𝑋)) |
18 | 15, 17 | eqbrtrd 5169 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ (♯‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 0cc0 11106 ℤcz 12554 ♯chash 14286 ∥ cdvds 16193 Basecbs 17140 Grpcgrp 18815 .gcmg 18944 SubGrpcsubg 18994 odcod 19386 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-eqg 18999 df-od 19390 |
This theorem is referenced by: odsubdvds 19433 gexcl2 19451 gexdvds3 19452 pgpfi1 19457 prmcyg 19756 lt6abl 19757 ablfacrp 19930 pgpfac1lem2 19939 dchrfi 26747 dchrabs 26752 |
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