MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gexex Structured version   Visualization version   GIF version

Theorem gexex 18302
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 805 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel)
5 simplr 807 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ)
6 simprl 809 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥𝑋)
71, 3odf 18002 . . . . . . . 8 𝑂:𝑋⟶ℕ0
8 frn 6091 . . . . . . . 8 (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
97, 8ax-mp 5 . . . . . . 7 ran 𝑂 ⊆ ℕ0
10 nn0ssz 11436 . . . . . . 7 0 ⊆ ℤ
119, 10sstri 3645 . . . . . 6 ran 𝑂 ⊆ ℤ
1211a1i 11 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → ran 𝑂 ⊆ ℤ)
13 nnz 11437 . . . . . . . 8 (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
1413adantl 481 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
15 ablgrp 18244 . . . . . . . . . . . 12 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1615adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
171, 2, 3gexod 18047 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
1816, 17sylan 487 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
191, 3odcl 18001 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑂𝑥) ∈ ℕ0)
2019adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℕ0)
2120nn0zd 11518 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℤ)
22 simplr 807 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → 𝐸 ∈ ℕ)
23 dvdsle 15079 . . . . . . . . . . 11 (((𝑂𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2421, 22, 23syl2anc 694 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → ((𝑂𝑥) ∥ 𝐸 → (𝑂𝑥) ≤ 𝐸))
2518, 24mpd 15 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ≤ 𝐸)
2625ralrimiva 2995 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∀𝑥𝑋 (𝑂𝑥) ≤ 𝐸)
27 ffn 6083 . . . . . . . . . 10 (𝑂:𝑋⟶ℕ0𝑂 Fn 𝑋)
287, 27ax-mp 5 . . . . . . . . 9 𝑂 Fn 𝑋
29 breq1 4688 . . . . . . . . . 10 (𝑦 = (𝑂𝑥) → (𝑦𝐸 ↔ (𝑂𝑥) ≤ 𝐸))
3029ralrn 6402 . . . . . . . . 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 4689 . . . . . . . . 9 (𝑛 = 𝐸 → (𝑦𝑛𝑦𝐸))
3433ralbidv 3015 . . . . . . . 8 (𝑛 = 𝐸 → (∀𝑦 ∈ ran 𝑂 𝑦𝑛 ↔ ∀𝑦 ∈ ran 𝑂 𝑦𝐸))
3534rspcev 3340 . . . . . . 7 ((𝐸 ∈ ℤ ∧ ∀𝑦 ∈ ran 𝑂 𝑦𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3614, 32, 35syl2anc 694 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛)
3736ad2antrr 762 . . . . 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 6396 . . . . . 6 ((𝑂 Fn 𝑋𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
4038, 39sylan 487 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ∈ ran 𝑂)
41 suprzub 11817 . . . . 5 ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛 ∧ (𝑂𝑦) ∈ ran 𝑂) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
4212, 37, 40, 41syl3anc 1366 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
43 simplrr 818 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))
4442, 43breqtrrd 4713 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝑥))
451, 2, 3, 4, 5, 6, 44gexexlem 18301 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂𝑥) = 𝐸)
4611a1i 11 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ⊆ ℤ)
471grpbn0 17498 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4816, 47syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
497fdmi 6090 . . . . . . . 8 dom 𝑂 = 𝑋
5049eqeq1i 2656 . . . . . . 7 (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
51 dm0rn0 5374 . . . . . . 7 (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
5250, 51bitr3i 266 . . . . . 6 (𝑋 = ∅ ↔ ran 𝑂 = ∅)
5352necon3bii 2875 . . . . 5 (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
5448, 53sylib 208 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
55 suprzcl2 11816 . . . 4 ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5646, 54, 36, 55syl3anc 1366 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
57 fvelrnb 6282 . . . 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 3047 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   class class class wbr 4685  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  supcsup 8387  cr 9973   < clt 10112  cle 10113  cn 11058  0cn0 11330  cz 11415  cdvds 15027  Basecbs 15904  Grpcgrp 17469  odcod 17990  gExcgex 17991  Abelcabl 18240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-od 17994  df-gex 17995  df-cmn 18241  df-abl 18242
This theorem is referenced by:  cyggexb  18346  pgpfaclem3  18528
  Copyright terms: Public domain W3C validator