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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 17778 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑂‘𝑋) ∈ ℕ0) |
7 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ℕ0) |
8 | 3, 7 | eqeltrrd 2689 | . . . . . 6 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ0) |
9 | fvex 6113 | . . . . . . . 8 ⊢ (Base‘𝐺) ∈ V | |
10 | 4, 9 | eqeltri 2684 | . . . . . . 7 ⊢ 𝐵 ∈ V |
11 | hashclb 13011 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ0)) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ0) |
13 | 8, 12 | sylibr 223 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin) |
14 | eqid 2610 | . . . . . 6 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
15 | eqid 2610 | . . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} | |
16 | 4, 14, 15, 5 | cyggenod 18109 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (#‘𝐵)))) |
17 | 1, 13, 16 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (#‘𝐵)))) |
18 | 2, 3, 17 | mpbir2and 959 | . . 3 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵}) |
19 | ne0i 3880 | . . 3 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅) |
21 | 4, 14, 15 | iscyg2 18107 | . 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵} ≠ ∅)) |
22 | 1, 20, 21 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℕ0cn0 11169 ℤcz 11254 #chash 12979 Basecbs 15695 Grpcgrp 17245 .gcmg 17363 odcod 17767 CycGrpccyg 18102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-od 17771 df-cyg 18103 |
This theorem is referenced by: prmcyg 18118 lt6abl 18119 |
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