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Theorem cygabl 18557
Description: A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
cygabl (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Proof of Theorem cygabl
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2764 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2764 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg3 18553 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)))
4 eqidd 2765 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5 eqidd 2765 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (+g𝐺) = (+g𝐺))
6 simpll 783 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Grp)
7 eqeq1 2768 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑛(.g𝐺)𝑥)))
87rexbidv 3198 . . . . . . . . 9 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥)))
9 oveq1 6848 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑛(.g𝐺)𝑥) = (𝑚(.g𝐺)𝑥))
109eqeq2d 2774 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑎 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑚(.g𝐺)𝑥)))
1110cbvrexv 3319 . . . . . . . . 9 (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥))
128, 11syl6bb 278 . . . . . . . 8 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1312rspccv 3457 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1413adantl 473 . . . . . 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 2768 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑏 = (𝑛(.g𝐺)𝑥)))
1615rexbidv 3198 . . . . . . . 8 (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1716rspccv 3457 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1817adantl 473 . . . . . 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 3253 . . . . . . . 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 11628 . . . . . . . . . . . . . 14 (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
2120ad2antrl 719 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22 zcn 11628 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
2322ad2antll 720 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
2421, 23addcomd 10491 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
2524oveq1d 6856 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑛 + 𝑚)(.g𝐺)𝑥))
26 simpll 783 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27 simprl 787 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28 simprr 789 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29 simplr 785 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑥 ∈ (Base‘𝐺))
30 eqid 2764 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
311, 2, 30mulgdir 17839 . . . . . . . . . . . 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 1491 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
331, 2, 30mulgdir 17839 . . . . . . . . . . . 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 1491 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑛 + 𝑚)(.g𝐺)𝑥) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3525, 32, 343eqtr3d 2806 . . . . . . . . . 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 6850 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
37 oveq12 6850 . . . . . . . . . . . 12 ((𝑏 = (𝑛(.g𝐺)𝑥) ∧ 𝑎 = (𝑚(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3837ancoms 450 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3936, 38eqeq12d 2779 . . . . . . . . . 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 238 . . . . . . . . 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 3184 . . . . . . . 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 234 . . . . . . 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 472 . . . . . 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 601 . . . . 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 1144 . . . 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 18471 . . 3 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
4746r19.29an 3223 . 2 ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
483, 47sylbi 208 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3054  wrex 3055  cfv 6067  (class class class)co 6841  cc 10186   + caddc 10191  cz 11623  Basecbs 16131  +gcplusg 16215  Grpcgrp 17690  .gcmg 17808  Abelcabl 18459  CycGrpccyg 18544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-n0 11538  df-z 11624  df-uz 11886  df-fz 12533  df-seq 13008  df-0g 16369  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-grp 17693  df-minusg 17694  df-mulg 17809  df-cmn 18460  df-abl 18461  df-cyg 18545
This theorem is referenced by:  lt6abl  18561  frgpcyg  20193
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