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Theorem cyggenod 18332
 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.)
Hypotheses
Ref Expression
iscyg.1 𝐵 = (Base‘𝐺)
iscyg.2 · = (.g𝐺)
iscyg3.e 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
cyggenod.o 𝑂 = (od‘𝐺)
Assertion
Ref Expression
cyggenod ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋𝐸 ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (#‘𝐵))))
Distinct variable groups:   𝑥,𝑛,𝐵   𝑛,𝑂   𝑛,𝑋,𝑥   𝑛,𝐺,𝑥   · ,𝑛,𝑥
Allowed substitution hints:   𝐸(𝑥,𝑛)   𝑂(𝑥)

Proof of Theorem cyggenod
StepHypRef Expression
1 iscyg.1 . . 3 𝐵 = (Base‘𝐺)
2 iscyg.2 . . 3 · = (.g𝐺)
3 iscyg3.e . . 3 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
41, 2, 3iscyggen 18328 . 2 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
5 simplr 807 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → 𝐵 ∈ Fin)
6 simplll 813 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → 𝐺 ∈ Grp)
7 simpr 476 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
8 simplr 807 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → 𝑋𝐵)
91, 2mulgcl 17606 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
106, 7, 8, 9syl3anc 1366 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑋) ∈ 𝐵)
11 eqid 2651 . . . . . . . 8 (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))
1210, 11fmptd 6425 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵)
13 frn 6091 . . . . . . 7 ((𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵)
1412, 13syl 17 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵)
15 ssfi 8221 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin)
165, 14, 15syl2anc 694 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin)
17 hashen 13175 . . . . 5 ((ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (#‘𝐵) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵))
1816, 5, 17syl2anc 694 . . . 4 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ((#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (#‘𝐵) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵))
19 cyggenod.o . . . . . . . 8 𝑂 = (od‘𝐺)
201, 19, 2, 11dfod2 18027 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑂𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0))
2120adantlr 751 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (𝑂𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0))
2216iftrued 4127 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0) = (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))))
2321, 22eqtr2d 2686 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (𝑂𝑋))
2423eqeq1d 2653 . . . 4 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ((#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (#‘𝐵) ↔ (𝑂𝑋) = (#‘𝐵)))
25 fisseneq 8212 . . . . . . 7 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)
26253expia 1286 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
27 enrefg 8029 . . . . . . . 8 (𝐵 ∈ Fin → 𝐵𝐵)
2827adantr 480 . . . . . . 7 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → 𝐵𝐵)
29 breq1 4688 . . . . . . 7 (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵𝐵𝐵))
3028, 29syl5ibrcom 237 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵))
3126, 30impbid 202 . . . . 5 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
325, 14, 31syl2anc 694 . . . 4 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
3318, 24, 323bitr3rd 299 . . 3 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ (𝑂𝑋) = (#‘𝐵)))
3433pm5.32da 674 . 2 ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → ((𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (#‘𝐵))))
354, 34syl5bb 272 1 ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋𝐸 ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (#‘𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {crab 2945   ⊆ wss 3607  ifcif 4119   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ≈ cen 7994  Fincfn 7997  0cc0 9974  ℤcz 11415  #chash 13157  Basecbs 15904  Grpcgrp 17469  .gcmg 17587  odcod 17990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-od 17994 This theorem is referenced by:  iscygodd  18336  cyggexb  18346
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