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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 19682 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (♯‘𝐵)) |
4 | 3 | expcom 415 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
5 | 4 | adantl 483 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
6 | simpll 766 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Abel) | |
7 | ablgrp 19574 | . . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | 7 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Grp) |
9 | simplr 768 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐵 ∈ Fin) | |
10 | 1, 2 | gexcl2 19378 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ) |
11 | 8, 9, 10 | syl2anc 585 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐸 ∈ ℕ) |
12 | eqid 2737 | . . . . . 6 ⊢ (od‘𝐺) = (od‘𝐺) | |
13 | 1, 2, 12 | gexex 19638 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
14 | 6, 11, 13 | syl2anc 585 | . . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
15 | simplr 768 | . . . . . . 7 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → 𝐸 = (♯‘𝐵)) | |
16 | 15 | eqeq2d 2748 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 ↔ ((od‘𝐺)‘𝑥) = (♯‘𝐵))) |
17 | eqid 2737 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
18 | eqid 2737 | . . . . . . . . . 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 19668 | . . . . . . . . 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 585 | . . . . . . . 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 4299 | . . . . . . . . 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 19666 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
23 | 22 | baib 537 | . . . . . . . . . 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 246 | . . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} → 𝐺 ∈ CycGrp)) |
26 | 20, 25 | sylbird 260 | . . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → ((𝑥 ∈ 𝐵 ∧ ((od‘𝐺)‘𝑥) = (♯‘𝐵)) → 𝐺 ∈ CycGrp)) |
27 | 26 | expdimp 454 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = (♯‘𝐵) → 𝐺 ∈ CycGrp)) |
28 | 16, 27 | sylbid 239 | . . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 → 𝐺 ∈ CycGrp)) |
29 | 28 | rexlimdva 3153 | . . . 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 414 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐸 = (♯‘𝐵) → 𝐺 ∈ CycGrp)) |
32 | 5, 31 | impbid 211 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (♯‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 {crab 3410 ∅c0 4287 ↦ cmpt 5193 ran crn 5639 ‘cfv 6501 (class class class)co 7362 Fincfn 8890 ℕcn 12160 ℤcz 12506 ♯chash 14237 Basecbs 17090 Grpcgrp 18755 .gcmg 18879 odcod 19313 gExcgex 19314 Abelcabl 19570 CycGrpccyg 19661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-dvds 16144 df-gcd 16382 df-prm 16555 df-pc 16716 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-eqg 18934 df-od 19317 df-gex 19318 df-cmn 19571 df-abl 19572 df-cyg 19662 |
This theorem is referenced by: (None) |
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