Theorem gexex 19826

Description: In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if 𝐸 = 0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so 𝐸 is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
gexex.1	⊢ 𝑋 = (Base‘𝐺)
gexex.2	⊢ 𝐸 = (gEx‘𝐺)
gexex.3	⊢ 𝑂 = (od‘𝐺)

Assertion

Ref	Expression
gexex	⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸)

Distinct variable groups:   𝑥,𝐸   𝑥,𝐺   𝑥,𝑂   𝑥,𝑋

Proof of Theorem gexex

Dummy variables 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexex.1	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		gexex.2	. . 3 ⊢ 𝐸 = (gEx‘𝐺)
3		gexex.3	. . 3 ⊢ 𝑂 = (od‘𝐺)
4		simpll 772	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5		simplr 774	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6		simprl 776	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥 ∈ 𝑋)
7	1, 3	odf 19510	. . . . . . 7 ⊢ 𝑂:𝑋⟶ℕ0
8		frn 6669	. . . . . . 7 ⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ran 𝑂 ⊆ ℕ0
10		nn0ssz 12545	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
11	9, 10	sstri 3931	. . . . 5 ⊢ ran 𝑂 ⊆ ℤ
12		nnz 12543	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
13	12	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14		ablgrp 19758	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
15	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
16	1, 2, 3	gexod 19559	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
17	15, 16	sylan 586	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
18	1, 3	odcl 19509	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0)
19	18	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ0)
20	19	nn0zd 12547	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
21		simplr 774	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ)
22		dvdsle 16277	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸))
23	20, 21, 22	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸))
24	17, 23	mpd 15	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ≤ 𝐸)
25	24	ralrimiva 3132	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)
26		ffn 6662	. . . . . . . . . 10 ⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋)
27	7, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝑂 Fn 𝑋
28		breq1 5082	. . . . . . . . . 10 ⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸))
29	28	ralrn 7036	. . . . . . . . 9 ⊢ (𝑂 Fn 𝑋 → (∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸))
30	27, 29	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)
31	25, 30	sylibr 235	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸)
32		brralrspcev 5139	. . . . . . 7 ⊢ ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
33	13, 31, 32	syl2anc 590	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
34	33	ad2antrr 732	. . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
35	27	a1i 11	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋)
36		fnfvelrn 7028	. . . . . 6 ⊢ ((𝑂 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂)
37	35, 36	sylan 586	. . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂)
38		suprzub 12887	. . . . 5 ⊢ ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛 ∧ (𝑂‘𝑦) ∈ ran 𝑂) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < ))
39	11, 34, 37, 38	mp3an2i 1474	. . . 4 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < ))
40		simplrr 783	. . . 4 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))
41	39, 40	breqtrrd 5107	. . 3 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝑥))
42	1, 2, 3, 4, 5, 6, 41	gexexlem 19825	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸)
43	1	grpbn0 18940	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
44	15, 43	syl 17	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
45	7	fdmi 6673	. . . . . . . 8 ⊢ dom 𝑂 = 𝑋
46	45	eqeq1i 2745	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47		dm0rn0 5873	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
48	46, 47	bitr3i 278	. . . . . 6 ⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅)
49	48	necon3bii 2987	. . . . 5 ⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
50	44, 49	sylib 219	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51		suprzcl2 12886	. . . 4 ⊢ ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
52	11, 50, 33, 51	mp3an2i 1474	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53		fvelrnb 6894	. . . 4 ⊢ (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )))
54	27, 53	ax-mp 5	. . 3 ⊢ (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))
55	52, 54	sylib 219	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))
56	42, 55	reximddv 3156	1 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079  dom cdm 5625  ran crn 5626   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  supcsup 9350  ℝcr 11035   < clt 11177   ≤ cle 11178  ℕcn 12172  ℕ₀cn0 12435  ℤcz 12522   ∥ cdvds 16219  Basecbs 17177  Grpcgrp 18907  odcod 19497  gExcgex 19498  Abelcabl 19754

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-q 12897  df-rp 12941  df-fz 13460  df-fzo 13607  df-fl 13749  df-mod 13827  df-seq 13962  df-exp 14022  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-dvds 16220  df-gcd 16462  df-prm 16639  df-pc 16806  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-sbg 18912  df-mulg 19042  df-od 19501  df-gex 19502  df-cmn 19755  df-abl 19756

This theorem is referenced by:  cyggexb  19872  pgpfaclem3  20058