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Theorem gexex 18237
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 789 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5 simplr 791 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6 simprl 793 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥𝑋)
71, 3odf 17937 . . . . . . . 8 𝑂:𝑋⟶ℕ0
8 frn 6040 . . . . . . . 8 (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
97, 8ax-mp 5 . . . . . . 7 ran 𝑂 ⊆ ℕ0
10 nn0ssz 11383 . . . . . . 7 0 ⊆ ℤ
119, 10sstri 3604 . . . . . 6 ran 𝑂 ⊆ ℤ
1211a1i 11 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ran 𝑂 ⊆ ℤ)
13 nnz 11384 . . . . . . . 8 (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
1413adantl 482 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
15 ablgrp 18179 . . . . . . . . . . . 12 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1615adantr 481 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
171, 2, 3gexod 17982 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
1816, 17sylan 488 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
191, 3odcl 17936 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑂𝑥) ∈ ℕ0)
2019adantl 482 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℕ0)
2120nn0zd 11465 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℤ)
22 simplr 791 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → 𝐸 ∈ ℕ)
23 dvdsle 15013 . . . . . . . . . . 11 (((𝑂𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2421, 22, 23syl2anc 692 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2518, 24mpd 15 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ≤ 𝐸)
2625ralrimiva 2963 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
27 ffn 6032 . . . . . . . . . 10 (𝑂:𝑋⟶ℕ0𝑂 Fn 𝑋)
287, 27ax-mp 5 . . . . . . . . 9 𝑂 Fn 𝑋
29 breq1 4647 . . . . . . . . . 10 (𝑦 = (𝑂𝑥) → (𝑦𝐸 ↔ (𝑂𝑥) ≤ 𝐸))
3029ralrn 6348 . . . . . . . . 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 224 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑦 ∈ ran 𝑂 𝑦𝐸)
33 breq2 4648 . . . . . . . . 9 (𝑛 = 𝐸 → (𝑦𝑛𝑦𝐸))
3433ralbidv 2983 . . . . . . . 8 (𝑛 = 𝐸 → (∀𝑦 ∈ ran 𝑂 𝑦𝑛 ↔ ∀𝑦 ∈ ran 𝑂 𝑦𝐸))
3534rspcev 3304 . . . . . . 7 ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3614, 32, 35syl2anc 692 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3736ad2antrr 761 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3828a1i 11 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋)
39 fnfvelrn 6342 . . . . . 6 ((𝑂 Fn 𝑋𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
4038, 39sylan 488 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
41 suprzub 11764 . . . . 5 ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛 ∧ (𝑂𝑦) ∈ ran 𝑂) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
4212, 37, 40, 41syl3anc 1324 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
43 simplrr 800 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
4442, 43breqtrrd 4672 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝑥))
451, 2, 3, 4, 5, 6, 44gexexlem 18236 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂𝑥) = 𝐸)
4611a1i 11 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ⊆ ℤ)
471grpbn0 17432 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4816, 47syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
497fdmi 6039 . . . . . . . 8 dom 𝑂 = 𝑋
5049eqeq1i 2625 . . . . . . 7 (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
51 dm0rn0 5331 . . . . . . 7 (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
5250, 51bitr3i 266 . . . . . 6 (𝑋 = ∅ ↔ ran 𝑂 = ∅)
5352necon3bii 2843 . . . . 5 (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
5448, 53sylib 208 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
55 suprzcl2 11763 . . . 4 ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5646, 54, 36, 55syl3anc 1324 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
57 fvelrnb 6230 . . . 4 (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < )))
5828, 57ax-mp 5 . . 3 (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
5956, 58sylib 208 . 2 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
6045, 59reximddv 3015 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  wss 3567  c0 3907   class class class wbr 4644  dom cdm 5104  ran crn 5105   Fn wfn 5871  wf 5872  cfv 5876  supcsup 8331  cr 9920   < clt 10059  cle 10060  cn 11005  0cn0 11277  cz 11362  cdvds 14964  Basecbs 15838  Grpcgrp 17403  odcod 17925  gExcgex 17926  Abelcabl 18175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-dvds 14965  df-gcd 15198  df-prm 15367  df-pc 15523  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-sbg 17408  df-mulg 17522  df-od 17929  df-gex 17930  df-cmn 18176  df-abl 18177
This theorem is referenced by:  cyggexb  18281  pgpfaclem3  18463
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