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Mirrors > Home > MPE Home > Th. List > cyggenod | Structured version Visualization version GIF version |
Description: An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
iscyg.1 | ⊢ 𝐵 = (Base‘𝐺) |
iscyg.2 | ⊢ · = (.g‘𝐺) |
iscyg3.e | ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
cyggenod.o | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
cyggenod | ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscyg.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | iscyg.2 | . . 3 ⊢ · = (.g‘𝐺) | |
3 | iscyg3.e | . . 3 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
4 | 1, 2, 3 | iscyggen 19468 | . 2 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
5 | simplr 766 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ Fin) | |
6 | simplll 772 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝐺 ∈ Grp) | |
7 | simpr 485 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
8 | simplr 766 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝑋 ∈ 𝐵) | |
9 | 1, 2 | mulgcl 18709 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑛 · 𝑋) ∈ 𝐵) |
10 | 6, 7, 8, 9 | syl3anc 1370 | . . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑋) ∈ 𝐵) |
11 | 10 | fmpttd 6982 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵) |
12 | 11 | frnd 6601 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) |
13 | 5, 12 | ssfid 9030 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin) |
14 | hashen 14049 | . . . . 5 ⊢ ((ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (♯‘𝐵) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵)) | |
15 | 13, 5, 14 | syl2anc 584 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ((♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (♯‘𝐵) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵)) |
16 | cyggenod.o | . . . . . . . 8 ⊢ 𝑂 = (od‘𝐺) | |
17 | eqid 2738 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) | |
18 | 1, 16, 2, 17 | dfod2 19159 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑂‘𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0)) |
19 | 18 | adantlr 712 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (𝑂‘𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0)) |
20 | 13 | iftrued 4468 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0) = (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))) |
21 | 19, 20 | eqtr2d 2779 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (𝑂‘𝑋)) |
22 | 21 | eqeq1d 2740 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ((♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (♯‘𝐵) ↔ (𝑂‘𝑋) = (♯‘𝐵))) |
23 | fisseneq 9022 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) | |
24 | 23 | 3expia 1120 | . . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
25 | enrefg 8760 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → 𝐵 ≈ 𝐵) | |
26 | 25 | adantr 481 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → 𝐵 ≈ 𝐵) |
27 | breq1 5077 | . . . . . . 7 ⊢ (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ 𝐵 ≈ 𝐵)) | |
28 | 26, 27 | syl5ibrcom 246 | . . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵)) |
29 | 24, 28 | impbid 211 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
30 | 5, 12, 29 | syl2anc 584 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
31 | 15, 22, 30 | 3bitr3rd 310 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ (𝑂‘𝑋) = (♯‘𝐵))) |
32 | 31 | pm5.32da 579 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → ((𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
33 | 4, 32 | syl5bb 283 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 ifcif 4460 class class class wbr 5074 ↦ cmpt 5157 ran crn 5586 ‘cfv 6427 (class class class)co 7268 ≈ cen 8718 Fincfn 8721 0cc0 10859 ℤcz 12307 ♯chash 14032 Basecbs 16900 Grpcgrp 18565 .gcmg 18688 odcod 19120 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-oadd 8289 df-omul 8290 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9685 df-acn 9688 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-fz 13228 df-fl 13500 df-mod 13578 df-seq 13710 df-exp 13771 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-dvds 15952 df-0g 17140 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-grp 18568 df-minusg 18569 df-sbg 18570 df-mulg 18689 df-od 19124 |
This theorem is referenced by: iscygodd 19476 cyggexb 19488 |
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