![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
5 | ssidd 4001 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ⊆ 𝐵) | |
6 | 1, 3 | grpcl 18802 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
7 | eqid 2731 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 1, 7 | grpidcl 18825 | . 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
9 | eqid 2731 | . 2 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | 1, 9 | grpinvcl 18847 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mulgsubcl 18940 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 ℤcz 12540 Basecbs 17126 +gcplusg 17179 0gc0g 17367 Grpcgrp 18794 invgcminusg 18795 .gcmg 18922 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-seq 13949 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-mulg 18923 |
This theorem is referenced by: mulgneg 18944 mulgnegneg 18945 mulgcld 18948 mulgaddcomlem 18949 mulgaddcom 18950 mulginvcom 18951 mulgdirlem 18957 mulgdir 18958 mulgass 18963 mulgmodid 18965 mulgsubdir 18966 cycsubgcl 19049 ghmmulg 19070 odmod 19378 odcong 19381 odmulgid 19386 odmulg 19388 odmulgeq 19389 odbezout 19390 odf1 19394 dfod2 19396 odf1o2 19405 gexdvds 19416 mulgdi 19655 mulgghm 19657 mulgsubdi 19658 odadd2 19677 gexexlem 19680 iscyggen2 19708 cyggenod 19711 iscyg3 19713 ablfacrp 19895 pgpfac1lem2 19904 pgpfac1lem3a 19905 pgpfac1lem3 19906 pgpfac1lem4 19907 mulgass2 20078 mulgghm2 20979 mulgrhm 20980 zlmlmod 21009 cygznlem2a 21056 isarchi3 32204 archirng 32205 archirngz 32206 archiabllem1a 32208 archiabllem2c 32212 freshmansdream 32249 isarchiofld 32297 |
Copyright terms: Public domain | W3C validator |