Theorem cygablOLD 19407

Description: Obsolete proof of cygabl 19406 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 19401	. 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 763	. . . 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 3225	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥)))
9		oveq1 7262	. . . . . . . . . . 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 3378	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥))
12	8, 11	bitrdi 286	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
13	12	rspccv 3549	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g‘𝐺)𝑥)))
14	13	adantl 481	. . . . . 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 3225	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
17	16	rspccv 3549	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g‘𝐺)𝑥)))
18	17	adantl 481	. . . . . 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 3292	. . . . . . . 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 12254	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
21	20	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22		zcn 12254	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
23	22	ad2antll 725	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
24	21, 23	addcomd 11107	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
25	24	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑛 + 𝑚)(.g‘𝐺)𝑥))
26		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27		simprl 767	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28		simprr 769	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29		simplr 765	. . . . . . . . . . . 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 18650	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 18650	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 7264	. . . . . . . . . . 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 7264	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 3222	. . . . . . . 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 480	. . . . . 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 607	. . . . 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 1114	. . . 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 19315	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel)
47	46	r19.29an 3216	. 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 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ‘cfv 6418  (class class class)co 7255  ℂcc 10800   + caddc 10805  ℤcz 12249  Basecbs 16840  +_gcplusg 16888  Grpcgrp 18492  ._gcmg 18615  Abelcabl 19302  CycGrpccyg 19392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-mulg 18616  df-cmn 19303  df-abl 19304  df-cyg 19393

This theorem is referenced by: (None)