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Mirrors > Home > MPE Home > Th. List > iscygodd | Structured version Visualization version GIF version |
Description: Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.) |
Ref | Expression |
---|---|
iscygodd.1 | ⊢ 𝐵 = (Base‘𝐺) |
iscygodd.o | ⊢ 𝑂 = (od‘𝐺) |
iscygodd.3 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
iscygodd.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
iscygodd.5 | ⊢ (𝜑 → (𝑂‘𝑋) = (♯‘𝐵)) |
Ref | Expression |
---|---|
iscygodd | ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscygodd.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | iscygodd.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | iscygodd.5 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = (♯‘𝐵)) | |
4 | iscygodd.1 | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺) | |
5 | iscygodd.o | . . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺) | |
6 | 4, 5 | odcl 18426 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑂‘𝑋) ∈ ℕ0) |
7 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ℕ0) |
8 | 3, 7 | eqeltrrd 2867 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
9 | 4 | fvexi 6513 | . . . . . . 7 ⊢ 𝐵 ∈ V |
10 | hashclb 13534 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔ (♯‘𝐵) ∈ ℕ0)) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (𝐵 ∈ Fin ↔ (♯‘𝐵) ∈ ℕ0) |
12 | 8, 11 | sylibr 226 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin) |
13 | eqid 2778 | . . . . . 6 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | eqid 2778 | . . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} | |
15 | 4, 13, 14, 5 | cyggenod 18759 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
16 | 1, 12, 15 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
17 | 2, 3, 16 | mpbir2and 700 | . . 3 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵}) |
18 | 17 | ne0d 4187 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅) |
19 | 4, 13, 14 | iscyg2 18757 | . 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅)) |
20 | 1, 18, 19 | sylanbrc 575 | 1 ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 {crab 3092 Vcvv 3415 ∅c0 4178 ↦ cmpt 5008 ran crn 5408 ‘cfv 6188 (class class class)co 6976 Fincfn 8306 ℕ0cn0 11707 ℤcz 11793 ♯chash 13505 Basecbs 16339 Grpcgrp 17891 .gcmg 18011 odcod 18414 CycGrpccyg 18752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-omul 7910 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-acn 9165 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-dvds 15468 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-od 18418 df-cyg 18753 |
This theorem is referenced by: prmcyg 18768 lt6abl 18769 |
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