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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 19931 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (♯‘𝐵)) |
4 | 3 | expcom 413 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp → 𝐸 = (♯‘𝐵))) |
6 | simpll 767 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Abel) | |
7 | ablgrp 19818 | . . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
8 | 7 | ad2antrr 726 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐺 ∈ Grp) |
9 | simplr 769 | . . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐵 ∈ Fin) | |
10 | 1, 2 | gexcl2 19622 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ) |
11 | 8, 9, 10 | syl2anc 584 | . . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → 𝐸 ∈ ℕ) |
12 | eqid 2735 | . . . . . 6 ⊢ (od‘𝐺) = (od‘𝐺) | |
13 | 1, 2, 12 | gexex 19886 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
14 | 6, 11, 13 | syl2anc 584 | . . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) → ∃𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) = 𝐸) |
15 | simplr 769 | . . . . . . 7 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → 𝐸 = (♯‘𝐵)) | |
16 | 15 | eqeq2d 2746 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 𝐸 ↔ ((od‘𝐺)‘𝑥) = (♯‘𝐵))) |
17 | eqid 2735 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
18 | eqid 2735 | . . . . . . . . . 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 19917 | . . . . . . . . 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 584 | . . . . . . . 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 4347 | . . . . . . . . 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 19915 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑦 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵} ≠ ∅)) |
23 | 22 | baib 535 | . . . . . . . . . 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 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 452 | . . . . . 6 ⊢ ((((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) ∧ 𝐸 = (♯‘𝐵)) ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = (♯‘𝐵) → 𝐺 ∈ CycGrp)) |
28 | 16, 27 | sylbid 240 | . . . . 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 412 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐸 = (♯‘𝐵) → 𝐺 ∈ CycGrp)) |
32 | 5, 31 | impbid 212 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (♯‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 ∅c0 4339 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℕcn 12264 ℤcz 12611 ♯chash 14366 Basecbs 17245 Grpcgrp 18964 .gcmg 19098 odcod 19557 gExcgex 19558 Abelcabl 19814 CycGrpccyg 19910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-eqg 19156 df-od 19561 df-gex 19562 df-cmn 19815 df-abl 19816 df-cyg 19911 |
This theorem is referenced by: (None) |
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