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Mirrors > Home > MPE Home > Th. List > cyggexb | Structured version Visualization version GIF version |
Description: A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygctb.1 | ⊢ 𝐵 = (Base‘𝐺) |
cyggex.o | ⊢ 𝐸 = (gEx‘𝐺) |
Ref | Expression |
---|---|
cyggexb | ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (#‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygctb.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | cyggex.o | . . . . 5 ⊢ 𝐸 = (gEx‘𝐺) | |
3 | 1, 2 | cyggex 18299 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (#‘𝐵)) |
4 | 3 | expcom 451 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐺 ∈ CycGrp → 𝐸 = (#‘𝐵))) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp → 𝐸 = (#‘𝐵))) |
6 | simpll 790 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → 𝐺 ∈ Abel) | |
7 | ablgrp 18198 | . . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | 7 | ad2antrr 762 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → 𝐺 ∈ Grp) |
9 | simplr 792 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → 𝐵 ∈ Fin) | |
10 | 1, 2 | gexcl2 18004 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ) |
11 | 8, 9, 10 | syl2anc 693 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → 𝐸 ∈ ℕ) |
12 | eqid 2622 | . . . . . 6 ⊢ (od‘𝐺) = (od‘𝐺) | |
13 | 1, 2, 12 | gexex 18256 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
14 | 6, 11, 13 | syl2anc 693 | . . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
15 | simplr 792 | . . . . . . 7 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) ∧ 𝑥 ∈ 𝐵) → 𝐸 = (#‘𝐵)) | |
16 | 15 | eqeq2d 2632 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 ↔ ((od‘𝐺)‘𝑥) = (#‘𝐵))) |
17 | eqid 2622 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
18 | eqid 2622 | . . . . . . . . . 10 ⊢ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} = {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} | |
19 | 1, 17, 18, 12 | cyggenod 18286 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ↔ (𝑥 ∈ 𝐵 ∧ ((od‘𝐺)‘𝑥) = (#‘𝐵)))) |
20 | 8, 9, 19 | syl2anc 693 | . . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ↔ (𝑥 ∈ 𝐵 ∧ ((od‘𝐺)‘𝑥) = (#‘𝐵)))) |
21 | ne0i 3921 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} → {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅) | |
22 | 1, 17, 18 | iscyg2 18284 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
23 | 22 | baib 944 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ CycGrp ↔ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
24 | 8, 23 | syl 17 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → (𝐺 ∈ CycGrp ↔ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
25 | 21, 24 | syl5ibr 236 | . . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} → 𝐺 ∈ CycGrp)) |
26 | 20, 25 | sylbird 250 | . . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → ((𝑥 ∈ 𝐵 ∧ ((od‘𝐺)‘𝑥) = (#‘𝐵)) → 𝐺 ∈ CycGrp)) |
27 | 26 | expdimp 453 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = (#‘𝐵) → 𝐺 ∈ CycGrp)) |
28 | 16, 27 | sylbid 230 | . . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 → 𝐺 ∈ CycGrp)) |
29 | 28 | rexlimdva 3031 | . . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → (∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸 → 𝐺 ∈ CycGrp)) |
30 | 14, 29 | mpd 15 | . . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (#‘𝐵)) → 𝐺 ∈ CycGrp) |
31 | 30 | ex 450 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐸 = (#‘𝐵) → 𝐺 ∈ CycGrp)) |
32 | 5, 31 | impbid 202 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (#‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ∅c0 3915 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℕcn 11020 ℤcz 11377 #chash 13117 Basecbs 15857 Grpcgrp 17422 .gcmg 17540 odcod 17944 gExcgex 17945 Abelcabl 18194 CycGrpccyg 18279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-od 17948 df-gex 17949 df-cmn 18195 df-abl 18196 df-cyg 18280 |
This theorem is referenced by: (None) |
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