Theorem gexex 19766

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 764	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5		simplr 766	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6		simprl 768	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥 ∈ 𝑋)
7	1, 3	odf 19450	. . . . . . 7 ⊢ 𝑂:𝑋⟶ℕ0
8		frn 6724	. . . . . . 7 ⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ ran 𝑂 ⊆ ℕ0
10		nn0ssz 12588	. . . . . 6 ⊢ ℕ0 ⊆ ℤ
11	9, 10	sstri 3991	. . . . 5 ⊢ ran 𝑂 ⊆ ℤ
12		nnz 12586	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14		ablgrp 19698	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
16	1, 2, 3	gexod 19499	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
17	15, 16	sylan 579	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸)
18	1, 3	odcl 19449	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ0)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ0)
20	19	nn0zd 12591	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
21		simplr 766	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ)
22		dvdsle 16260	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸))
23	20, 21, 22	syl2anc 583	. . . . . . . . . 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 3145	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)
26		ffn 6717	. . . . . . . . . 10 ⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋)
27	7, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝑂 Fn 𝑋
28		breq1 5151	. . . . . . . . . 10 ⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸))
29	28	ralrn 7089	. . . . . . . . 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 233	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸)
32		brralrspcev 5208	. . . . . . 7 ⊢ ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
33	13, 31, 32	syl2anc 583	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛)
34	33	ad2antrr 723	. . . . 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 579	. . . . 5 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂)
38		suprzub 12930	. . . . 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 1465	. . . 4 ⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < ))
40		simplrr 775	. . . 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 5176	. . 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 19765	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸)
43	1	grpbn0 18891	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
44	15, 43	syl 17	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
45	7	fdmi 6729	. . . . . . . 8 ⊢ dom 𝑂 = 𝑋
46	45	eqeq1i 2736	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47		dm0rn0 5924	. . . . . . 7 ⊢ (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
48	46, 47	bitr3i 277	. . . . . 6 ⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅)
49	48	necon3bii 2992	. . . . 5 ⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
50	44, 49	sylib 217	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51		suprzcl2 12929	. . . 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 1465	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53		fvelrnb 6952	. . . 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 217	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))
56	42, 55	reximddv 3170	1 ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148  dom cdm 5676  ran crn 5677   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  supcsup 9441  ℝcr 11115   < clt 11255   ≤ cle 11256  ℕcn 12219  ℕ₀cn0 12479  ℤcz 12565   ∥ cdvds 16204  Basecbs 17151  Grpcgrp 18858  odcod 19437  gExcgex 19438  Abelcabl 19694

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-fz 13492  df-fzo 13635  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-dvds 16205  df-gcd 16443  df-prm 16616  df-pc 16777  df-0g 17394  df-mgm 18568  df-sgrp 18647  df-mnd 18663  df-grp 18861  df-minusg 18862  df-sbg 18863  df-mulg 18991  df-od 19441  df-gex 19442  df-cmn 19695  df-abl 19696

This theorem is referenced by:  cyggexb  19812  pgpfaclem3  19998