Theorem gexex 20047

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 19731	. . . . . . 7 ⊢ 𝑂:𝑋⟶ℕ0
8		frn 6709	. . . . . . 7 ⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ran 𝑂 ⊆ ℕ0
10		nn0ssz 12697	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
11	9, 10	sstri 3940	. . . . 5 ⊢ ran 𝑂 ⊆ ℤ
12		nnz 12695	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
13	12	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14		ablgrp 19979	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
15	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
16	1, 2, 3	gexod 19780	. . . . . . . . . . 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 19730	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0)
19	18	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ0)
20	19	nn0zd 12699	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
21		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ)
22		dvdsle 16460	. . . . . . . . . . 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 3155	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)
26		ffn 6701	. . . . . . . . . 10 ⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋)
27	7, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝑂 Fn 𝑋
28		breq1 5106	. . . . . . . . . 10 ⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸))
29	28	ralrn 7080	. . . . . . . . 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 5165	. . . . . . 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 7072	. . . . . 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 13047	. . . . 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 5133	. . 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 20046	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸)
43	1	grpbn0 19157	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
44	15, 43	syl 18	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
45	7	fdmi 6713	. . . . . . . 8 ⊢ dom 𝑂 = 𝑋
46	45	eqeq1i 2766	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47		dm0rn0 5906	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
48	46, 47	bitr3i 280	. . . . . 6 ⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅)
49	48	necon3bii 3008	. . . . 5 ⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
50	44, 49	sylib 221	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51		suprzcl2 13046	. . . 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 6937	. . . 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 3179	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 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  dom cdm 5651  ran crn 5652   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  supcsup 9416  ℝcr 11180   < clt 11324   ≤ cle 11325  ℕcn 12316  ℕ₀cn0 12587  ℤcz 12674   ∥ cdvds 16402  Basecbs 17367  Grpcgrp 19124  odcod 19718  gExcgex 19719  Abelcabl 19975

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-mod 13990  df-seq 14125  df-exp 14185  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-dvds 16403  df-gcd 16645  df-prm 16827  df-pc 16995  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-minusg 19128  df-sbg 19129  df-mulg 19258  df-od 19722  df-gex 19723  df-cmn 19976  df-abl 19977

This theorem is used by:  cyggexb  20093  pgpfaclem3  20279