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Theorem gexex 19885
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 19569 . . . . . . 7 𝑂:𝑋⟶ℕ0
8 frn 6743 . . . . . . 7 (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆ ℕ0)
97, 8ax-mp 5 . . . . . 6 ran 𝑂 ⊆ ℕ0
10 nn0ssz 12633 . . . . . 6 0 ⊆ ℤ
119, 10sstri 4004 . . . . 5 ran 𝑂 ⊆ ℤ
12 nnz 12631 . . . . . . . 8 (𝐸 ∈ ℕ → 𝐸 ∈ ℤ)
1312adantl 481 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈ ℤ)
14 ablgrp 19817 . . . . . . . . . . . 12 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1514adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp)
161, 2, 3gexod 19618 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
1715, 16sylan 580 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∥ 𝐸)
181, 3odcl 19568 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑂𝑥) ∈ ℕ0)
1918adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℕ0)
2019nn0zd 12636 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → (𝑂𝑥) ∈ ℤ)
21 simplr 769 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥𝑋) → 𝐸 ∈ ℕ)
22 dvdsle 16343 . . . . . . . . . . 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 3143 . . . . . . . 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 5150 . . . . . . . . . 10 (𝑦 = (𝑂𝑥) → (𝑦𝐸 ↔ (𝑂𝑥) ≤ 𝐸))
2928ralrn 7107 . . . . . . . . 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 5207 . . . . . . 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 7099 . . . . . 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 12978 . . . . 5 ((ran 𝑂 ⊆ ℤ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛 ∧ (𝑂𝑦) ∈ ran 𝑂) → (𝑂𝑦) ≤ sup(ran 𝑂, ℝ, < ))
3911, 34, 37, 38mp3an2i 1465 . . . 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 5175 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝑥))
421, 2, 3, 4, 5, 6, 41gexexlem 19884 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥𝑋 ∧ (𝑂𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂𝑥) = 𝐸)
431grpbn0 18996 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4415, 43syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅)
457fdmi 6747 . . . . . . . 8 dom 𝑂 = 𝑋
4645eqeq1i 2739 . . . . . . 7 (dom 𝑂 = ∅ ↔ 𝑋 = ∅)
47 dm0rn0 5937 . . . . . . 7 (dom 𝑂 = ∅ ↔ ran 𝑂 = ∅)
4846, 47bitr3i 277 . . . . . 6 (𝑋 = ∅ ↔ ran 𝑂 = ∅)
4948necon3bii 2990 . . . . 5 (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅)
5044, 49sylib 218 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran 𝑂 ≠ ∅)
51 suprzcl2 12977 . . . 4 ((ran 𝑂 ⊆ ℤ ∧ ran 𝑂 ≠ ∅ ∧ ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5211, 50, 33, 51mp3an2i 1465 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53 fvelrnb 6968 . . . 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 3168 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  wss 3962  c0 4338   class class class wbr 5147  dom cdm 5688  ran crn 5689   Fn wfn 6557  wf 6558  cfv 6562  supcsup 9477  cr 11151   < clt 11292  cle 11293  cn 12263  0cn0 12523  cz 12610  cdvds 16286  Basecbs 17244  Grpcgrp 18963  odcod 19556  gExcgex 19557  Abelcabl 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-gcd 16528  df-prm 16705  df-pc 16870  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-od 19560  df-gex 19561  df-cmn 19814  df-abl 19815
This theorem is referenced by:  cyggexb  19931  pgpfaclem3  20117
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