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Mirrors > Home > MPE Home > Th. List > gexod | Structured version Visualization version GIF version |
Description: Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
gexod.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexod.2 | ⊢ 𝐸 = (gEx‘𝐺) |
gexod.3 | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
gexod | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gexod.1 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | gexod.2 | . . . 4 ⊢ 𝐸 = (gEx‘𝐺) | |
3 | eqid 2732 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | eqid 2732 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 1, 2, 3, 4 | gexid 19490 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐸(.g‘𝐺)𝐴) = (0g‘𝐺)) |
6 | 5 | adantl 482 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐸(.g‘𝐺)𝐴) = (0g‘𝐺)) |
7 | 1, 2 | gexcl 19489 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐸 ∈ ℕ0) |
8 | 7 | adantr 481 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐸 ∈ ℕ0) |
9 | 8 | nn0zd 12588 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐸 ∈ ℤ) |
10 | gexod.3 | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
11 | 1, 10, 3, 4 | oddvds 19456 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ ℤ) → ((𝑂‘𝐴) ∥ 𝐸 ↔ (𝐸(.g‘𝐺)𝐴) = (0g‘𝐺))) |
12 | 9, 11 | mpd3an3 1462 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) ∥ 𝐸 ↔ (𝐸(.g‘𝐺)𝐴) = (0g‘𝐺))) |
13 | 6, 12 | mpbird 256 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℕ0cn0 12476 ℤcz 12562 ∥ cdvds 16201 Basecbs 17148 0gc0g 17389 Grpcgrp 18855 .gcmg 18986 odcod 19433 gExcgex 19434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16202 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-od 19437 df-gex 19438 |
This theorem is referenced by: gexnnod 19497 gexexlem 19761 gexex 19762 cyggex2 19806 |
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