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Theorem cyggenod 19804
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 19800 . 2 (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡))
5 simplr 766 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ 𝐡 ∈ Fin)
6 simplll 772 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝐺 ∈ Grp)
7 simpr 484 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝑛 ∈ β„€)
8 simplr 766 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) ∧ 𝑛 ∈ β„€) β†’ 𝑋 ∈ 𝐡)
91, 2mulgcl 19018 . . . . . . . . 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 7110 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)):β„€βŸΆπ΅)
1211frnd 6719 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡)
135, 12ssfid 9269 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) ∈ Fin)
14 hashen 14312 . . . . 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 583 . . . 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 2726 . . . . . . . 8 (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))
181, 16, 2, 17dfod2 19484 . . . . . . 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 712 . . . . . 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 4531 . . . . . 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 2767 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (β™―β€˜ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋))) = (π‘‚β€˜π‘‹))
2221eqeq1d 2728 . . . 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 9259 . . . . . . 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 8982 . . . . . . . 8 (𝐡 ∈ Fin β†’ 𝐡 β‰ˆ 𝐡)
2625adantr 480 . . . . . . 7 ((𝐡 ∈ Fin ∧ ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) βŠ† 𝐡) β†’ 𝐡 β‰ˆ 𝐡)
27 breq1 5144 . . . . . . 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 583 . . . 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 310 . . 3 (((𝐺 ∈ Grp ∧ 𝐡 ∈ Fin) ∧ 𝑋 ∈ 𝐡) β†’ (ran (𝑛 ∈ β„€ ↦ (𝑛 Β· 𝑋)) = 𝐡 ↔ (π‘‚β€˜π‘‹) = (β™―β€˜π΅)))
3231pm5.32da 578 . 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 283 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 395   = wceq 1533   ∈ wcel 2098  {crab 3426   βŠ† wss 3943  ifcif 4523   class class class wbr 5141   ↦ cmpt 5224  ran crn 5670  β€˜cfv 6537  (class class class)co 7405   β‰ˆ cen 8938  Fincfn 8941  0cc0 11112  β„€cz 12562  β™―chash 14295  Basecbs 17153  Grpcgrp 18863  .gcmg 18995  odcod 19444
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-omul 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-fl 13763  df-mod 13841  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16205  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-od 19448
This theorem is referenced by:  iscygodd  19808  cyggexb  19819
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