Theorem cygablOLD 19011

Description: Obsolete proof of cygabl 19010 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 2821	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2821	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg3 19005	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)))
4		eqidd 2822	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5		eqidd 2822	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (+g‘𝐺) = (+g‘𝐺))
6		simpll 765	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Grp)
7		eqeq1 2825	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
8	7	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
9		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘𝐺)𝑥) = (𝑚(.g‘𝐺)𝑥))
10	9	eqeq2d 2832	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
11	10	cbvrexv 3453	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥))
12	8, 11	syl6bb 289	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
13	12	rspccv 3620	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
14	13	adantl 484	. . . . . 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 2825	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
16	15	rexbidv 3297	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
17	16	rspccv 3620	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
18	17	adantl 484	. . . . . 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 3367	. . . . . . . 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 11987	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
21	20	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22		zcn 11987	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
23	22	ad2antll 727	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
24	21, 23	addcomd 10842	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
25	24	oveq1d 7171	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑛 + 𝑚)(.g‘𝐺)𝑥))
26		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑥 ∈ (Base‘𝐺))
30		eqid 2821	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
31	1, 2, 30	mulgdir 18259	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 18259	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 2864	. . . . . . . . . 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 7165	. . . . . . . . . . 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 7165	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 2837	. . . . . . . . . 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 249	. . . . . . . . 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 3294	. . . . . . . 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 245	. . . . . . 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 483	. . . . . 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 609	. . . . 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 1112	. . . 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 18920	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
47	46	r19.29an 3288	. 2 ⊢ ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
48	3, 47	sylbi 219	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ‘cfv 6355  (class class class)co 7156  ℂcc 10535   + caddc 10540  ℤcz 11982  Basecbs 16483  +_gcplusg 16565  Grpcgrp 18103  ._gcmg 18224  Abelcabl 18907  CycGrpccyg 18996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-seq 13371  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-mulg 18225  df-cmn 18908  df-abl 18909  df-cyg 18997

This theorem is referenced by: (None)