Theorem cygablOLD 19490

Description: Obsolete proof of cygabl 19489 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.

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2740	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg3 19484	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)))
4		eqidd 2741	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5		eqidd 2741	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (+g‘𝐺) = (+g‘𝐺))
6		simpll 764	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Grp)
7		eqeq1 2744	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
8	7	rexbidv 3228	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
9		oveq1 7278	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
10	9	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
11	10	cbvrexv 3387	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥))
12	8, 11	bitrdi 287	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
13	12	rspccv 3558	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
14	13	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
15		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
16	15	rexbidv 3228	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
17	16	rspccv 3558	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
18	17	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
19		reeanv 3295	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) ↔ (∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
20		zcn 12324	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
21	20	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22		zcn 12324	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
23	22	ad2antll 726	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
24	21, 23	addcomd 11177	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
25	24	oveq1d 7286	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑛 + 𝑚)(.g‘𝐺)𝑥))
26		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27		simprl 768	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28		simprr 770	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑥 ∈ (Base‘𝐺))
30		eqid 2740	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
31	1, 2, 30	mulgdir 18733	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥)))
32	26, 27, 28, 29, 31	syl13anc 1371	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥)))
33	1, 2, 30	mulgdir 18733	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑛 + 𝑚)(.g‘𝐺)𝑥) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥)))
34	26, 28, 27, 29, 33	syl13anc 1371	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑛 + 𝑚)(.g‘𝐺)𝑥) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥)))
35	25, 32, 34	3eqtr3d 2788	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥)) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥)))
36		oveq12 7280	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑎(+g‘𝐺)𝑏) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥)))
37		oveq12 7280	. . . . . . . . . . . 12 ⊢ ((𝑏 = (𝑛(.g‘𝐺)𝑥) ∧ 𝑎 = (𝑚(.g‘𝐺)𝑥)) → (𝑏(+g‘𝐺)𝑎) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥)))
38	37	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑏(+g‘𝐺)𝑎) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥)))
39	36, 38	eqeq12d 2756	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → ((𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎) ↔ ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥)) = ((𝑛(.g‘𝐺)𝑥)(+g‘𝐺)(𝑚(.g‘𝐺)𝑥))))
40	35, 39	syl5ibrcom 246	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)))
41	40	rexlimdvva 3225	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)))
42	19, 41	syl5bir 242	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)))
43	42	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)))
44	14, 18, 43	syl2and 608	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → ((𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)))
45	44	3impib 1115	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
46	4, 5, 6, 45	isabld 19398	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
47	46	r19.29an 3219	. 2 ⊢ ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
48	3, 47	sylbi 216	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  ‘cfv 6432  (class class class)co 7271  ℂcc 10870   + caddc 10875  ℤcz 12319  Basecbs 16910  +_gcplusg 16960  Grpcgrp 18575  ._gcmg 18698  Abelcabl 19385  CycGrpccyg 19475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2374  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-seq 13720  df-0g 17150  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-grp 18578  df-minusg 18579  df-mulg 18699  df-cmn 19386  df-abl 19387  df-cyg 19476

This theorem is referenced by: (None)