Theorem cygablOLD 19492

Description: Obsolete proof of cygabl 19491 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 2738	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2738	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg3 19486	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)))
4		eqidd 2739	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5		eqidd 2739	. . . 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 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
8	7	rexbidv 3226	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
9		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
10	9	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
11	10	cbvrexv 3389	. . . . . . . . 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 2742	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
16	15	rexbidv 3226	. . . . . . . 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 3294	. . . . . . . 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 7290	. . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
31	1, 2, 30	mulgdir 18735	. . . . . . . . . . . 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 18735	. . . . . . . . . . . 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 2786	. . . . . . . . . 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 7284	. . . . . . . . . . 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 7284	. . . . . . . . . . . 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 2754	. . . . . . . . . 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 3223	. . . . . . . 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 19400	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
47	46	r19.29an 3217	. 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 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ‘cfv 6433  (class class class)co 7275  ℂcc 10869   + caddc 10874  ℤcz 12319  Basecbs 16912  +_gcplusg 16962  Grpcgrp 18577  ._gcmg 18700  Abelcabl 19387  CycGrpccyg 19477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-seq 13722  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-mulg 18701  df-cmn 19388  df-abl 19389  df-cyg 19478

This theorem is referenced by: (None)