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Theorem gexex 19871
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) → ∃𝑥𝑋 (𝑂𝑥) = 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.
StepHypRef Expression
1 gexex.1 . . 3 𝑋 = (Base‘𝐺)
2 gexex.2 . . 3 𝐸 = (gEx‘𝐺)
3 gexex.3 . . 3 𝑂 = (od‘𝐺)
4 simpll 767 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5 simplr 769 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6 simprl 771 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥𝑋)
71, 3odf 19555 . . . . . . 7 𝑂:𝑋⟶ℕ0
8 frn 6743 . . . . . . 7 (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
97, 8ax-mp 5 . . . . . 6 ran 𝑂 ⊆ ℕ0
10 nn0ssz 12636 . . . . . 6 0 ⊆ ℤ
119, 10sstri 3993 . . . . 5 ran 𝑂 ⊆ ℤ
12 nnz 12634 . . . . . . . 8 (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
1312adantl 481 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14 ablgrp 19803 . . . . . . . . . . . 12 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1514adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
161, 2, 3gexod 19604 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
1715, 16sylan 580 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
181, 3odcl 19554 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑂𝑥) ∈ ℕ0)
1918adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℕ0)
2019nn0zd 12639 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℤ)
21 simplr 769 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → 𝐸 ∈ ℕ)
22 dvdsle 16347 . . . . . . . . . . 11 (((𝑂𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2320, 21, 22syl2anc 584 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2417, 23mpd 15 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ≤ 𝐸)
2524ralrimiva 3146 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
26 ffn 6736 . . . . . . . . . 10 (𝑂:𝑋⟶ℕ0𝑂 Fn 𝑋)
277, 26ax-mp 5 . . . . . . . . 9 𝑂 Fn 𝑋
28 breq1 5146 . . . . . . . . . 10 (𝑦 = (𝑂𝑥) → (𝑦𝐸 ↔ (𝑂𝑥) ≤ 𝐸))
2928ralrn 7108 . . . . . . . . 9 (𝑂 Fn 𝑋 → (∀𝑦 ∈ ran 𝑂 𝑦𝐸 ↔ ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸))
3027, 29ax-mp 5 . . . . . . . 8 (∀𝑦 ∈ ran 𝑂 𝑦𝐸 ↔ ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
3125, 30sylibr 234 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦𝐸)
32 brralrspcev 5203 . . . . . . 7 ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3313, 31, 32syl2anc 584 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3433ad2antrr 726 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3527a1i 11 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋)
36 fnfvelrn 7100 . . . . . 6 ((𝑂 Fn 𝑋𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
3735, 36sylan 580 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
38 suprzub 12981 . . . . 5 ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛 ∧ (𝑂𝑦) ∈ ran 𝑂) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
3911, 34, 37, 38mp3an2i 1468 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
40 simplrr 778 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
4139, 40breqtrrd 5171 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝑥))
421, 2, 3, 4, 5, 6, 41gexexlem 19870 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂𝑥) = 𝐸)
431grpbn0 18984 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4415, 43syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
457fdmi 6747 . . . . . . . 8 dom 𝑂 = 𝑋
4645eqeq1i 2742 . . . . . . 7 (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47 dm0rn0 5935 . . . . . . 7 (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
4846, 47bitr3i 277 . . . . . 6 (𝑋 = ∅ ↔ ran 𝑂 = ∅)
4948necon3bii 2993 . . . . 5 (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
5044, 49sylib 218 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51 suprzcl2 12980 . . . 4 ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5211, 50, 33, 51mp3an2i 1468 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53 fvelrnb 6969 . . . 4 (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < )))
5427, 53ax-mp 5 . . 3 (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
5552, 54sylib 218 . 2 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
5642, 55reximddv 3171 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333   class class class wbr 5143  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  supcsup 9480  cr 11154   < clt 11295  cle 11296  cn 12266  0cn0 12526  cz 12613  cdvds 16290  Basecbs 17247  Grpcgrp 18951  odcod 19542  gExcgex 19543  Abelcabl 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-od 19546  df-gex 19547  df-cmn 19800  df-abl 19801
This theorem is referenced by:  cyggexb  19917  pgpfaclem3  20103
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