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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 2736 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | eqid 2736 | . . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) | |
5 | 1, 2, 3, 4 | dfod2 18909 | . . . 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 1133 | . . 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 1139 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ Fin) | |
8 | 1, 3, 4 | cycsubgcl 18567 | . . . . . . . 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 1133 | . . . . . . 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 498 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ (SubGrp‘𝐺)) |
11 | 1 | subgss 18498 | . . . . . 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 8876 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)) ∈ Fin) |
14 | 13 | iftrued 4433 | . . 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 2771 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)))) |
16 | 1 | lagsubg 18560 | . . 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 587 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (♯‘ran (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴))) ∥ (♯‘𝑋)) |
18 | 15, 17 | eqbrtrd 5061 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ (♯‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ifcif 4425 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 0cc0 10694 ℤcz 12141 ♯chash 13861 ∥ cdvds 15778 Basecbs 16666 Grpcgrp 18319 .gcmg 18442 SubGrpcsubg 18491 odcod 18870 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 df-dvds 15779 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-eqg 18496 df-od 18874 |
This theorem is referenced by: odsubdvds 18914 gexcl2 18932 gexdvds3 18933 pgpfi1 18938 prmcyg 19233 lt6abl 19234 ablfacrp 19407 pgpfac1lem2 19416 dchrfi 26090 dchrabs 26095 |
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