Theorem gexex 19954

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 779	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5		simplr 781	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6		simprl 783	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥 ∈ 𝑋)
7	1, 3	odf 19638	. . . . . . 7 ⊢ 𝑂:𝑋⟶ℕ0
8		frn 6720	. . . . . . 7 ⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ran 𝑂 ⊆ ℕ0
10		nn0ssz 12632	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
11	9, 10	sstri 3949	. . . . 5 ⊢ ran 𝑂 ⊆ ℤ
12		nnz 12630	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
13	12	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14		ablgrp 19886	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
15	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
16	1, 2, 3	gexod 19687	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
17	15, 16	sylan 592	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
18	1, 3	odcl 19637	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0)
19	18	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ0)
20	19	nn0zd 12634	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
21		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ)
22		dvdsle 16393	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸))
23	20, 21, 22	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸))
24	17, 23	mpd 16	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ≤ 𝐸)
25	24	ralrimiva 3160	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)
26		ffn 6712	. . . . . . . . . 10 ⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋)
27	7, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝑂 Fn 𝑋
28		breq1 5117	. . . . . . . . . 10 ⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸))
29	28	ralrn 7090	. . . . . . . . 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 237	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸)
32		brralrspcev 5176	. . . . . . 7 ⊢ ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
33	13, 31, 32	syl2anc 596	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
34	33	ad2antrr 739	. . . . 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 7082	. . . . . 6 ⊢ ((𝑂 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂)
37	35, 36	sylan 592	. . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂)
38		suprzub 12981	. . . . 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 1495	. . . 4 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < ))
40		simplrr 790	. . . 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 5144	. . 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 19953	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸)
43	1	grpbn0 19064	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
44	15, 43	syl 18	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
45	7	fdmi 6724	. . . . . . . 8 ⊢ dom 𝑂 = 𝑋
46	45	eqeq1i 2771	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47		dm0rn0 5919	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
48	46, 47	bitr3i 280	. . . . . 6 ⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅)
49	48	necon3bii 3013	. . . . 5 ⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
50	44, 49	sylib 221	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51		suprzcl2 12980	. . . 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 1495	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53		fvelrnb 6948	. . . 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 221	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))
56	42, 55	reximddv 3184	1 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908  ∅c0 4289   class class class wbr 5114  dom cdm 5666  ran crn 5667   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  supcsup 9410  ℝcr 11117   < clt 11261   ≤ cle 11262  ℕcn 12251  ℕ₀cn0 12522  ℤcz 12609   ∥ cdvds 16335  Basecbs 17294  Grpcgrp 19031  odcod 19625  gExcgex 19626  Abelcabl 19882

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-q 12991  df-rp 13035  df-fz 13554  df-fzo 13702  df-fl 13845  df-mod 13923  df-seq 14058  df-exp 14118  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-dvds 16336  df-gcd 16578  df-prm 16755  df-pc 16922  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035  df-sbg 19036  df-mulg 19165  df-od 19629  df-gex 19630  df-cmn 19883  df-abl 19884

This theorem is used by:  cyggexb  20000  pgpfaclem3  20186