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Theorem cygablOLD 19002
Description: Obsolete proof of cygabl 19001 as of 20-Jan-2024. A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
cygablOLD (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Proof of Theorem cygablOLD
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2822 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg3 18996 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)))
4 eqidd 2823 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5 eqidd 2823 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (+g𝐺) = (+g𝐺))
6 simpll 766 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Grp)
7 eqeq1 2826 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑛(.g𝐺)𝑥)))
87rexbidv 3283 . . . . . . . . 9 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥)))
9 oveq1 7147 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑛(.g𝐺)𝑥) = (𝑚(.g𝐺)𝑥))
109eqeq2d 2833 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑎 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑚(.g𝐺)𝑥)))
1110cbvrexv 3428 . . . . . . . . 9 (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥))
128, 11syl6bb 290 . . . . . . . 8 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1312rspccv 3595 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1413adantl 485 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
15 eqeq1 2826 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑏 = (𝑛(.g𝐺)𝑥)))
1615rexbidv 3283 . . . . . . . 8 (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1716rspccv 3595 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1817adantl 485 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
19 reeanv 3348 . . . . . . . 8 (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) ↔ (∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
20 zcn 11974 . . . . . . . . . . . . . 14 (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
2120ad2antrl 727 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22 zcn 11974 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
2322ad2antll 728 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
2421, 23addcomd 10831 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
2524oveq1d 7155 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑛 + 𝑚)(.g𝐺)𝑥))
26 simpll 766 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27 simprl 770 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28 simprr 772 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29 simplr 768 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑥 ∈ (Base‘𝐺))
30 eqid 2822 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
311, 2, 30mulgdir 18250 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
3226, 27, 28, 29, 31syl13anc 1369 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
331, 2, 30mulgdir 18250 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑛 + 𝑚)(.g𝐺)𝑥) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3426, 28, 27, 29, 33syl13anc 1369 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑛 + 𝑚)(.g𝐺)𝑥) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3525, 32, 343eqtr3d 2865 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
36 oveq12 7149 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
37 oveq12 7149 . . . . . . . . . . . 12 ((𝑏 = (𝑛(.g𝐺)𝑥) ∧ 𝑎 = (𝑚(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3837ancoms 462 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3936, 38eqeq12d 2838 . . . . . . . . . 10 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → ((𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎) ↔ ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥))))
4035, 39syl5ibrcom 250 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4140rexlimdvva 3280 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4219, 41syl5bir 246 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4342adantr 484 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4414, 18, 43syl2and 610 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ((𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
45443impib 1113 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
464, 5, 6, 45isabld 18911 . . 3 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
4746r19.29an 3274 . 2 ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
483, 47sylbi 220 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cfv 6334  (class class class)co 7140  cc 10524   + caddc 10529  cz 11969  Basecbs 16474  +gcplusg 16556  Grpcgrp 18094  .gcmg 18215  Abelcabl 18898  CycGrpccyg 18987
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  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
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-seq 13365  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-mulg 18216  df-cmn 18899  df-abl 18900  df-cyg 18988
This theorem is referenced by: (None)
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