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Theorem gexex 19638
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 = (Baseβ€˜πΊ)
gexex.2 𝐸 = (gExβ€˜πΊ)
gexex.3 𝑂 = (odβ€˜πΊ)
Assertion
Ref Expression
gexex ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆƒπ‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) = 𝐸)
Distinct variable groups:   π‘₯,𝐸   π‘₯,𝐺   π‘₯,𝑂   π‘₯,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 = (Baseβ€˜πΊ)
2 gexex.2 . . 3 𝐸 = (gExβ€˜πΊ)
3 gexex.3 . . 3 𝑂 = (odβ€˜πΊ)
4 simpll 766 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) β†’ 𝐺 ∈ Abel)
5 simplr 768 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) β†’ 𝐸 ∈ β„•)
6 simprl 770 . . 3 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) β†’ π‘₯ ∈ 𝑋)
71, 3odf 19326 . . . . . . 7 𝑂:π‘‹βŸΆβ„•0
8 frn 6680 . . . . . . 7 (𝑂:π‘‹βŸΆβ„•0 β†’ ran 𝑂 βŠ† β„•0)
97, 8ax-mp 5 . . . . . 6 ran 𝑂 βŠ† β„•0
10 nn0ssz 12529 . . . . . 6 β„•0 βŠ† β„€
119, 10sstri 3958 . . . . 5 ran 𝑂 βŠ† β„€
12 nnz 12527 . . . . . . . 8 (𝐸 ∈ β„• β†’ 𝐸 ∈ β„€)
1312adantl 483 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ 𝐸 ∈ β„€)
14 ablgrp 19574 . . . . . . . . . . . 12 (𝐺 ∈ Abel β†’ 𝐺 ∈ Grp)
1514adantr 482 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ 𝐺 ∈ Grp)
161, 2, 3gexod 19375 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ π‘₯ ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) βˆ₯ 𝐸)
1715, 16sylan 581 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) βˆ₯ 𝐸)
181, 3odcl 19325 . . . . . . . . . . . . 13 (π‘₯ ∈ 𝑋 β†’ (π‘‚β€˜π‘₯) ∈ β„•0)
1918adantl 483 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) ∈ β„•0)
2019nn0zd 12532 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) ∈ β„€)
21 simplr 768 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ 𝐸 ∈ β„•)
22 dvdsle 16199 . . . . . . . . . . 11 (((π‘‚β€˜π‘₯) ∈ β„€ ∧ 𝐸 ∈ β„•) β†’ ((π‘‚β€˜π‘₯) βˆ₯ 𝐸 β†’ (π‘‚β€˜π‘₯) ≀ 𝐸))
2320, 21, 22syl2anc 585 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ ((π‘‚β€˜π‘₯) βˆ₯ 𝐸 β†’ (π‘‚β€˜π‘₯) ≀ 𝐸))
2417, 23mpd 15 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ π‘₯ ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) ≀ 𝐸)
2524ralrimiva 3144 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆ€π‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) ≀ 𝐸)
26 ffn 6673 . . . . . . . . . 10 (𝑂:π‘‹βŸΆβ„•0 β†’ 𝑂 Fn 𝑋)
277, 26ax-mp 5 . . . . . . . . 9 𝑂 Fn 𝑋
28 breq1 5113 . . . . . . . . . 10 (𝑦 = (π‘‚β€˜π‘₯) β†’ (𝑦 ≀ 𝐸 ↔ (π‘‚β€˜π‘₯) ≀ 𝐸))
2928ralrn 7043 . . . . . . . . 9 (𝑂 Fn 𝑋 β†’ (βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝐸 ↔ βˆ€π‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) ≀ 𝐸))
3027, 29ax-mp 5 . . . . . . . 8 (βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝐸 ↔ βˆ€π‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) ≀ 𝐸)
3125, 30sylibr 233 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝐸)
32 brralrspcev 5170 . . . . . . 7 ((𝐸 ∈ β„€ ∧ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝐸) β†’ βˆƒπ‘› ∈ β„€ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝑛)
3313, 31, 32syl2anc 585 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆƒπ‘› ∈ β„€ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝑛)
3433ad2antrr 725 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) β†’ βˆƒπ‘› ∈ β„€ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝑛)
3527a1i 11 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) β†’ 𝑂 Fn 𝑋)
36 fnfvelrn 7036 . . . . . 6 ((𝑂 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘‚β€˜π‘¦) ∈ ran 𝑂)
3735, 36sylan 581 . . . . 5 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) β†’ (π‘‚β€˜π‘¦) ∈ ran 𝑂)
38 suprzub 12871 . . . . 5 ((ran 𝑂 βŠ† β„€ ∧ βˆƒπ‘› ∈ β„€ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝑛 ∧ (π‘‚β€˜π‘¦) ∈ ran 𝑂) β†’ (π‘‚β€˜π‘¦) ≀ sup(ran 𝑂, ℝ, < ))
3911, 34, 37, 38mp3an2i 1467 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) β†’ (π‘‚β€˜π‘¦) ≀ sup(ran 𝑂, ℝ, < ))
40 simplrr 777 . . . 4 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) β†’ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))
4139, 40breqtrrd 5138 . . 3 ((((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) β†’ (π‘‚β€˜π‘¦) ≀ (π‘‚β€˜π‘₯))
421, 2, 3, 4, 5, 6, 41gexexlem 19637 . 2 (((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) ∧ (π‘₯ ∈ 𝑋 ∧ (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))) β†’ (π‘‚β€˜π‘₯) = 𝐸)
431grpbn0 18786 . . . . . 6 (𝐺 ∈ Grp β†’ 𝑋 β‰  βˆ…)
4415, 43syl 17 . . . . 5 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ 𝑋 β‰  βˆ…)
457fdmi 6685 . . . . . . . 8 dom 𝑂 = 𝑋
4645eqeq1i 2742 . . . . . . 7 (dom 𝑂 = βˆ… ↔ 𝑋 = βˆ…)
47 dm0rn0 5885 . . . . . . 7 (dom 𝑂 = βˆ… ↔ ran 𝑂 = βˆ…)
4846, 47bitr3i 277 . . . . . 6 (𝑋 = βˆ… ↔ ran 𝑂 = βˆ…)
4948necon3bii 2997 . . . . 5 (𝑋 β‰  βˆ… ↔ ran 𝑂 β‰  βˆ…)
5044, 49sylib 217 . . . 4 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ ran 𝑂 β‰  βˆ…)
51 suprzcl2 12870 . . . 4 ((ran 𝑂 βŠ† β„€ ∧ ran 𝑂 β‰  βˆ… ∧ βˆƒπ‘› ∈ β„€ βˆ€π‘¦ ∈ ran 𝑂 𝑦 ≀ 𝑛) β†’ sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
5211, 50, 33, 51mp3an2i 1467 . . 3 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂)
53 fvelrnb 6908 . . . 4 (𝑂 Fn 𝑋 β†’ (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ βˆƒπ‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < )))
5427, 53ax-mp 5 . . 3 (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ βˆƒπ‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))
5552, 54sylib 217 . 2 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆƒπ‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) = sup(ran 𝑂, ℝ, < ))
5642, 55reximddv 3169 1 ((𝐺 ∈ Abel ∧ 𝐸 ∈ β„•) β†’ βˆƒπ‘₯ ∈ 𝑋 (π‘‚β€˜π‘₯) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074   βŠ† wss 3915  βˆ…c0 4287   class class class wbr 5110  dom cdm 5638  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  supcsup 9383  β„cr 11057   < clt 11196   ≀ cle 11197  β„•cn 12160  β„•0cn0 12420  β„€cz 12506   βˆ₯ cdvds 16143  Basecbs 17090  Grpcgrp 18755  odcod 19313  gExcgex 19314  Abelcabl 19570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-fz 13432  df-fzo 13575  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-dvds 16144  df-gcd 16382  df-prm 16555  df-pc 16716  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-mulg 18880  df-od 19317  df-gex 19318  df-cmn 19571  df-abl 19572
This theorem is referenced by:  cyggexb  19683  pgpfaclem3  19869
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