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Theorem cyggenod 18109
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 18105 . 2 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
5 simplr 788 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → 𝐵 ∈ Fin)
6 simplll 794 . . . . . . . . 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 788 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → 𝑋𝐵)
91, 2mulgcl 17382 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
106, 7, 8, 9syl3anc 1318 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑋) ∈ 𝐵)
11 eqid 2610 . . . . . . . 8 (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))
1210, 11fmptd 6292 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)):ℤ⟶𝐵)
13 frn 5966 . . . . . . 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 8065 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin)
165, 14, 15syl2anc 691 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin)
17 hashen 12997 . . . . 5 ((ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (#‘𝐵) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵))
1816, 5, 17syl2anc 691 . . . 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 17804 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑂𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0))
2120adantlr 747 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (𝑂𝑋) = if(ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ∈ Fin, (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))), 0))
2216iftrued 4044 . . . . . 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 2645 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (𝑂𝑋))
2423eqeq1d 2612 . . . 4 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → ((#‘ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) = (#‘𝐵) ↔ (𝑂𝑋) = (#‘𝐵)))
25 fisseneq 8056 . . . . . . 7 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)
26253expia 1259 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
27 enrefg 7873 . . . . . . . 8 (𝐵 ∈ Fin → 𝐵𝐵)
2827adantr 480 . . . . . . 7 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → 𝐵𝐵)
29 breq1 4586 . . . . . . 7 (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵𝐵𝐵))
3028, 29syl5ibrcom 236 . . . . . 6 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵))
3126, 30impbid 201 . . . . 5 ((𝐵 ∈ Fin ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ⊆ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
325, 14, 31syl2anc 691 . . . 4 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) ≈ 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
3318, 24, 323bitr3rd 298 . . 3 (((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) ∧ 𝑋𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵 ↔ (𝑂𝑋) = (#‘𝐵)))
3433pm5.32da 671 . 2 ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → ((𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵) ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (#‘𝐵))))
354, 34syl5bb 271 1 ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋𝐸 ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (#‘𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {crab 2900  wss 3540  ifcif 4036   class class class wbr 4583  cmpt 4643  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  cen 7838  Fincfn 7841  0cc0 9815  cz 11254  #chash 12979  Basecbs 15695  Grpcgrp 17245  .gcmg 17363  odcod 17767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-od 17771
This theorem is referenced by:  iscygodd  18113  cyggexb  18123
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