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Theorem gexex 18522
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 783 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5 simplr 785 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6 simprl 787 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥𝑋)
71, 3odf 18220 . . . . . . . 8 𝑂:𝑋⟶ℕ0
8 frn 6229 . . . . . . . 8 (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
97, 8ax-mp 5 . . . . . . 7 ran 𝑂 ⊆ ℕ0
10 nn0ssz 11645 . . . . . . 7 0 ⊆ ℤ
119, 10sstri 3770 . . . . . 6 ran 𝑂 ⊆ ℤ
1211a1i 11 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ran 𝑂 ⊆ ℤ)
13 nnz 11646 . . . . . . . 8 (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
1413adantl 473 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
15 ablgrp 18464 . . . . . . . . . . . 12 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1615adantr 472 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
171, 2, 3gexod 18265 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
1816, 17sylan 575 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
191, 3odcl 18219 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑂𝑥) ∈ ℕ0)
2019adantl 473 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℕ0)
2120nn0zd 11727 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℤ)
22 simplr 785 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → 𝐸 ∈ ℕ)
23 dvdsle 15317 . . . . . . . . . . 11 (((𝑂𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2421, 22, 23syl2anc 579 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2518, 24mpd 15 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ≤ 𝐸)
2625ralrimiva 3113 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
27 ffn 6223 . . . . . . . . . 10 (𝑂:𝑋⟶ℕ0𝑂 Fn 𝑋)
287, 27ax-mp 5 . . . . . . . . 9 𝑂 Fn 𝑋
29 breq1 4812 . . . . . . . . . 10 (𝑦 = (𝑂𝑥) → (𝑦𝐸 ↔ (𝑂𝑥) ≤ 𝐸))
3029ralrn 6552 . . . . . . . . 9 (𝑂 Fn 𝑋 → (∀𝑦 ∈ ran 𝑂 𝑦𝐸 ↔ ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸))
3128, 30ax-mp 5 . . . . . . . 8 (∀𝑦 ∈ ran 𝑂 𝑦𝐸 ↔ ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
3226, 31sylibr 225 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦𝐸)
33 brralrspcev 4869 . . . . . . 7 ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3414, 32, 33syl2anc 579 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3534ad2antrr 717 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3628a1i 11 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋)
37 fnfvelrn 6546 . . . . . 6 ((𝑂 Fn 𝑋𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
3836, 37sylan 575 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
39 suprzub 11980 . . . . 5 ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛 ∧ (𝑂𝑦) ∈ ran 𝑂) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
4012, 35, 38, 39syl3anc 1490 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
41 simplrr 796 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
4240, 41breqtrrd 4837 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝑥))
431, 2, 3, 4, 5, 6, 42gexexlem 18521 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂𝑥) = 𝐸)
4411a1i 11 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ⊆ ℤ)
451grpbn0 17718 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4616, 45syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
477fdmi 6233 . . . . . . . 8 dom 𝑂 = 𝑋
4847eqeq1i 2770 . . . . . . 7 (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
49 dm0rn0 5510 . . . . . . 7 (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
5048, 49bitr3i 268 . . . . . 6 (𝑋 = ∅ ↔ ran 𝑂 = ∅)
5150necon3bii 2989 . . . . 5 (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
5246, 51sylib 209 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
53 suprzcl2 11979 . . . 4 ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5444, 52, 34, 53syl3anc 1490 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
55 fvelrnb 6432 . . . 4 (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < )))
5628, 55ax-mp 5 . . 3 (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
5754, 56sylib 209 . 2 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
5843, 57reximddv 3164 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  wss 3732  c0 4079   class class class wbr 4809  dom cdm 5277  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  supcsup 8553  cr 10188   < clt 10328  cle 10329  cn 11274  0cn0 11538  cz 11624  cdvds 15265  Basecbs 16130  Grpcgrp 17689  odcod 18208  gExcgex 18209  Abelcabl 18460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-dvds 15266  df-gcd 15498  df-prm 15666  df-pc 15821  df-0g 16368  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-od 18212  df-gex 18213  df-cmn 18461  df-abl 18462
This theorem is referenced by:  cyggexb  18566  pgpfaclem3  18749
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