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Theorem cyggenod 19853
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 𝐡 = (Baseβ€˜πΊ)
iscyg.2 Β· = (.gβ€˜πΊ)
iscyg3.e 𝐸 = {π‘₯ ∈ 𝐡 ∣ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· π‘₯)) = 𝐡}
cyggenod.o 𝑂 = (odβ€˜πΊ)
Assertion
Ref Expression
cyggenod ((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) β†’ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ (π‘‚β€˜π‘‹) = (β™―β€˜π΅))))
Distinct variable groups:   π‘₯,𝑛,𝐡   𝑛,𝑂   𝑛,𝑋,π‘₯   𝑛,𝐺,π‘₯   Β· ,𝑛,π‘₯
Allowed substitution hints:   𝐸(π‘₯,𝑛)   𝑂(π‘₯)
Proof of Theorem cyggenod
StepHypRef Expression
1 iscyg.1 . . 3 𝐡 = (Baseβ€˜πΊ)
2 iscyg.2 . . 3 Β· = (.gβ€˜πΊ)
3 iscyg3.e . . 3 𝐸 = {π‘₯ ∈ 𝐡 ∣ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· π‘₯)) = 𝐡}
41, 2, 3iscyggen 19849 . 2 (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡))
5 simplr 767 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ 𝐡 ∈ Fin)
6 simplll 773 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝐺 ∈ Grp)
7 simpr 483 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝑛 ∈ β„€)
8 simplr 767 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝑋 ∈ 𝐡)
91, 2mulgcl 19060 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑛 ∈ β„€ ∧ 𝑋 ∈ 𝐡) β†’ (𝑛 Β· 𝑋) ∈ 𝐡)
106, 7, 8, 9syl3anc 1368 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ (𝑛 Β· 𝑋) ∈ 𝐡)
1110fmpttd 7130 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)):β„€βŸΆπ΅)
1211frnd 6735 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡)
135, 12ssfid 9300 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin)
14 hashen 14348 . . . . 5 ((ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))) = (β™―β€˜π΅) ↔ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡))
1513, 5, 14syl2anc 582 . . . 4 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ((β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))) = (β™―β€˜π΅) ↔ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡))
16 cyggenod.o . . . . . . . 8 𝑂 = (odβ€˜πΊ)
17 eqid 2728 . . . . . . . 8 (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))
181, 16, 2, 17dfod2 19533 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ (π‘‚β€˜π‘‹) = if(ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin, (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))), 0))
1918adantlr 713 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (π‘‚β€˜π‘‹) = if(ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin, (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))), 0))
2013iftrued 4540 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ if(ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin, (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))), 0) = (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))))
2119, 20eqtr2d 2769 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))) = (π‘‚β€˜π‘‹))
2221eqeq1d 2730 . . . 4 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ((β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))) = (β™―β€˜π΅) ↔ (π‘‚β€˜π‘‹) = (β™―β€˜π΅)))
23 fisseneq 9290 . . . . . . 7 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡 ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡) β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡)
24233expia 1118 . . . . . 6 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡 β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡))
25 enrefg 9013 . . . . . . . 8 (𝐡 ∈ Fin β†’ 𝐡 β‰ˆ 𝐡)
2625adantr 479 . . . . . . 7 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡) β†’ 𝐡 β‰ˆ 𝐡)
27 breq1 5155 . . . . . . 7 (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡 β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡 ↔ 𝐡 β‰ˆ 𝐡))
2826, 27syl5ibrcom 246 . . . . . 6 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡 β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡))
2924, 28impbid 211 . . . . 5 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡 ↔ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡))
305, 12, 29syl2anc 582 . . . 4 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) β‰ˆ 𝐡 ↔ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡))
3115, 22, 303bitr3rd 309 . . 3 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡 ↔ (π‘‚β€˜π‘‹) = (β™―β€˜π΅)))
3231pm5.32da 577 . 2 ((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) β†’ ((𝑋 ∈ 𝐡 ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡) ↔ (𝑋 ∈ 𝐡 ∧ (π‘‚β€˜π‘‹) = (β™―β€˜π΅))))
334, 32bitrid 282 1 ((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) β†’ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ (π‘‚β€˜π‘‹) = (β™―β€˜π΅))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430   βŠ† wss 3949  ifcif 4532   class class class wbr 5152   ↦ cmpt 5235  ran crn 5683  β€˜cfv 6553  (class class class)co 7426   β‰ˆ cen 8969  Fincfn 8972  0cc0 11148  β„€cz 12598  β™―chash 14331  Basecbs 17189  Grpcgrp 18904  .gcmg 19037  odcod 19493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-inf2 9674  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-oadd 8499  df-omul 8500  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-sup 9475  df-inf 9476  df-oi 9543  df-card 9972  df-acn 9975  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12253  df-2 12315  df-3 12316  df-n0 12513  df-z 12599  df-uz 12863  df-rp 13017  df-fz 13527  df-fl 13799  df-mod 13877  df-seq 14009  df-exp 14069  df-hash 14332  df-cj 15088  df-re 15089  df-im 15090  df-sqrt 15224  df-abs 15225  df-dvds 16241  df-0g 17432  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-grp 18907  df-minusg 18908  df-sbg 18909  df-mulg 19038  df-od 19497
This theorem is referenced by:  iscygodd  19857  cyggexb  19868
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