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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 18666 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑂‘𝑋) ∈ ℕ0) |
7 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ℕ0) |
8 | 3, 7 | eqeltrrd 2916 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
9 | 4 | fvexi 6686 | . . . . . . 7 ⊢ 𝐵 ∈ V |
10 | hashclb 13722 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔ (♯‘𝐵) ∈ ℕ0)) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (𝐵 ∈ Fin ↔ (♯‘𝐵) ∈ ℕ0) |
12 | 8, 11 | sylibr 236 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin) |
13 | eqid 2823 | . . . . . 6 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | eqid 2823 | . . . . . 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 19005 | . . . . 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 586 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
17 | 2, 3, 16 | mpbir2and 711 | . . 3 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵}) |
18 | 17 | ne0d 4303 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅) |
19 | 4, 13, 14 | iscyg2 19003 | . 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅)) |
20 | 1, 18, 19 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 ∅c0 4293 ↦ cmpt 5148 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℕ0cn0 11900 ℤcz 11984 ♯chash 13693 Basecbs 16485 Grpcgrp 18105 .gcmg 18226 odcod 18654 CycGrpccyg 18998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-od 18658 df-cyg 18999 |
This theorem is referenced by: prmcyg 19016 lt6abl 19017 |
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