![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18992 | . 2 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
5 | simplr 768 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ Fin) | |
6 | simplll 774 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝐺 ∈ Grp) | |
7 | simpr 488 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
8 | simplr 768 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → 𝑋 ∈ 𝐵) | |
9 | 1, 2 | mulgcl 18237 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑛 · 𝑋) ∈ 𝐵) |
10 | 6, 7, 8, 9 | syl3anc 1368 | . . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑋) ∈ 𝐵) |
11 | 10 | fmpttd 6856 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵) |
12 | 11 | frnd 6494 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) |
13 | 5, 12 | ssfid 8725 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin) |
14 | hashen 13703 | . . . . 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 587 | . . . 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 2798 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) | |
18 | 1, 16, 2, 17 | dfod2 18683 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑂‘𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0)) |
19 | 18 | adantlr 714 | . . . . . 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 4433 | . . . . . 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 2834 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (𝑂‘𝑋)) |
22 | 21 | eqeq1d 2800 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → ((♯‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (♯‘𝐵) ↔ (𝑂‘𝑋) = (♯‘𝐵))) |
23 | fisseneq 8713 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) | |
24 | 23 | 3expia 1118 | . . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
25 | enrefg 8524 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → 𝐵 ≈ 𝐵) | |
26 | 25 | adantr 484 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → 𝐵 ≈ 𝐵) |
27 | breq1 5033 | . . . . . . 7 ⊢ (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ 𝐵 ≈ 𝐵)) | |
28 | 26, 27 | syl5ibrcom 250 | . . . . . 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 215 | . . . . 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 587 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
31 | 15, 22, 30 | 3bitr3rd 313 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ (𝑂‘𝑋) = (♯‘𝐵))) |
32 | 31 | pm5.32da 582 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → ((𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
33 | 4, 32 | syl5bb 286 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ≈ cen 8489 Fincfn 8492 0cc0 10526 ℤcz 11969 ♯chash 13686 Basecbs 16475 Grpcgrp 18095 .gcmg 18216 odcod 18644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-od 18648 |
This theorem is referenced by: iscygodd 19000 cyggexb 19012 |
Copyright terms: Public domain | W3C validator |