![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19860 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (♯‘𝐵)) |
4 | 3 | expcom 412 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
5 | 4 | adantl 480 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
6 | simpll 765 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Abel) | |
7 | ablgrp 19747 | . . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | 7 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Grp) |
9 | simplr 767 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐵 ∈ Fin) | |
10 | 1, 2 | gexcl2 19551 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ) |
11 | 8, 9, 10 | syl2anc 582 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐸 ∈ ℕ) |
12 | eqid 2728 | . . . . . 6 ⊢ (od‘𝐺) = (od‘𝐺) | |
13 | 1, 2, 12 | gexex 19815 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
14 | 6, 11, 13 | syl2anc 582 | . . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
15 | simplr 767 | . . . . . . 7 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → 𝐸 = (♯‘𝐵)) | |
16 | 15 | eqeq2d 2739 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 ↔ ((od‘𝐺)‘𝑥) = (♯‘𝐵))) |
17 | eqid 2728 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
18 | eqid 2728 | . . . . . . . . . 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 19846 | . . . . . . . . 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 582 | . . . . . . . 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 4338 | . . . . . . . . 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 19844 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
23 | 22 | baib 534 | . . . . . . . . . 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 | imbitrrid 245 | . . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} → 𝐺 ∈ CycGrp)) |
26 | 20, 25 | sylbird 259 | . . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → ((𝑥 ∈ 𝐵 ∧ ((od‘𝐺)‘𝑥) = (♯‘𝐵)) → 𝐺 ∈ CycGrp)) |
27 | 26 | expdimp 451 | . . . . . 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 3152 | . . . 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 411 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 {crab 3430 ∅c0 4326 ↦ cmpt 5235 ran crn 5683 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 ℕcn 12250 ℤcz 12596 ♯chash 14329 Basecbs 17187 Grpcgrp 18897 .gcmg 19030 odcod 19486 gExcgex 19487 Abelcabl 19743 CycGrpccyg 19839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-dvds 16239 df-gcd 16477 df-prm 16650 df-pc 16813 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-eqg 19087 df-od 19490 df-gex 19491 df-cmn 19744 df-abl 19745 df-cyg 19840 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |