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Mirrors > Home > MPE Home > Th. List > mulgcl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnncl.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnncl.t | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulgcl | ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnncl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mulgnncl.t | . 2 ⊢ · = (.g‘𝐺) | |
3 | eqid 2821 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
5 | ssidd 3989 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ⊆ 𝐵) | |
6 | 1, 3 | grpcl 18051 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
7 | eqid 2821 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 1, 7 | grpidcl 18071 | . 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
9 | eqid 2821 | . 2 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | 1, 9 | grpinvcl 18091 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mulgsubcl 18182 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ℤcz 11970 Basecbs 16473 +gcplusg 16555 0gc0g 16703 Grpcgrp 18043 invgcminusg 18044 .gcmg 18164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-seq 13360 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-minusg 18047 df-mulg 18165 |
This theorem is referenced by: mulgneg 18186 mulgnegneg 18187 mulgcld 18189 mulgaddcomlem 18190 mulgaddcom 18191 mulginvcom 18192 mulgdirlem 18198 mulgdir 18199 mulgass 18204 mulgmodid 18206 mulgsubdir 18207 cycsubgcl 18289 ghmmulg 18310 odmod 18605 odcong 18608 odmulgid 18612 odmulg 18614 odmulgeq 18615 odbezout 18616 odf1 18620 dfod2 18622 odf1o2 18629 gexdvds 18640 mulgdi 18878 mulgghm 18880 mulgsubdi 18881 odadd2 18900 gexexlem 18903 iscyggen2 18931 cyggenod 18934 iscyg3 18936 ablfacrp 19119 pgpfac1lem2 19128 pgpfac1lem3a 19129 pgpfac1lem3 19130 pgpfac1lem4 19131 mulgass2 19282 mulgghm2 20574 mulgrhm 20575 zlmlmod 20600 cygznlem2a 20644 isarchi3 30744 archirng 30745 archirngz 30746 archiabllem1a 30748 archiabllem2c 30752 freshmansdream 30787 isarchiofld 30818 |
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