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Theorem cygabl 19924
Description: A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
Assertion
Ref Expression
cygabl (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Proof of Theorem cygabl
Dummy variables 𝑛 𝑥 𝑎 𝑏 𝑖 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2735 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg3 19919 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)))
4 eqidd 2736 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5 eqidd 2736 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (+g𝐺) = (+g𝐺))
6 simpll 767 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Grp)
7 oveq1 7438 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝑛(.g𝐺)𝑥) = (𝑖(.g𝐺)𝑥))
87eqeq2d 2746 . . . . . . . . . 10 (𝑛 = 𝑖 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑦 = (𝑖(.g𝐺)𝑥)))
98cbvrexvw 3236 . . . . . . . . 9 (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g𝐺)𝑥))
109biimpi 216 . . . . . . . 8 (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → ∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g𝐺)𝑥))
1110ralimi 3081 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g𝐺)𝑥))
1211adantl 481 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g𝐺)𝑥))
13123ad2ant1 1132 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g𝐺)𝑥))
14 simpll 767 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
15 simpr 484 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
1615anim1ci 616 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 ∈ (Base‘𝐺)))
17 df-3an 1088 . . . . . . . . . 10 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺)) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 ∈ (Base‘𝐺)))
1816, 17sylibr 234 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺)))
19 eqid 2735 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
201, 2, 19mulgdir 19137 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
2114, 18, 20syl2anc 584 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
2221ralrimivva 3200 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
2322adantr 480 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
24233ad2ant1 1132 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
25 simp2 1136 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
26 simp3 1137 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
27 zsscn 12619 . . . . . 6 ℤ ⊆ ℂ
2827a1i 11 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ℤ ⊆ ℂ)
2913, 24, 25, 26, 28cyccom 19234 . . . 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))
304, 5, 6, 29isabld 19828 . . 3 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
3130r19.29an 3156 . 2 ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
323, 31sylbi 217 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  cfv 6563  (class class class)co 7431  cc 11151   + caddc 11156  cz 12611  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  .gcmg 19098  Abelcabl 19814  CycGrpccyg 19910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-mulg 19099  df-cmn 19815  df-abl 19816  df-cyg 19911
This theorem is referenced by:  lt6abl  19928  frgpcyg  21610
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