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Mirrors > Home > MPE Home > Th. List > mulgnn0cl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnncl.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnncl.t | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulgnn0cl | ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnncl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mulgnncl.t | . 2 ⊢ · = (.g‘𝐺) | |
3 | eqid 2799 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mnd) | |
5 | ssidd 3820 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐵 ⊆ 𝐵) | |
6 | 1, 3 | mndcl 17616 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
7 | eqid 2799 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 1, 7 | mndidcl 17623 | . 2 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 17870 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ‘cfv 6101 (class class class)co 6878 ℕ0cn0 11580 Basecbs 16184 +gcplusg 16267 0gc0g 16415 Mndcmnd 17609 .gcmg 17856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-seq 13056 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mulg 17857 |
This theorem is referenced by: mulgnn0dir 17885 mulgnn0ass 17891 mhmmulg 17896 pwsmulg 17900 odmodnn0 18272 mulgmhm 18548 srgmulgass 18847 srgpcomp 18848 srgpcompp 18849 srgpcomppsc 18850 srgbinomlem1 18856 srgbinomlem2 18857 srgbinomlem4 18859 srgbinomlem 18860 lmodvsmmulgdi 19216 assamulgscmlem2 19672 mplcoe5lem 19790 mplcoe5 19791 psrbagev1 19832 evlslem3 19836 ply1moncl 19963 coe1pwmul 19971 ply1coefsupp 19987 ply1coe 19988 gsummoncoe1 19996 lply1binomsc 19999 evl1expd 20031 evl1scvarpw 20049 evl1scvarpwval 20050 evl1gsummon 20051 pmatcollpwscmatlem1 20922 mply1topmatcllem 20936 mply1topmatcl 20938 pm2mpghm 20949 monmat2matmon 20957 pm2mp 20958 chpscmatgsumbin 20977 chpscmatgsummon 20978 chfacfscmulcl 20990 chfacfscmul0 20991 chfacfpmmulcl 20994 chfacfpmmul0 20995 cpmadugsumlemB 21007 cpmadugsumlemC 21008 cpmadugsumlemF 21009 cayhamlem2 21017 cayhamlem4 21021 deg1pw 24221 plypf1 24309 lgsqrlem2 25424 lgsqrlem3 25425 lgsqrlem4 25426 omndmul2 30228 omndmul3 30229 omndmul 30230 isarchi2 30255 hbtlem4 38481 lmodvsmdi 42962 ply1mulgsumlem4 42976 ply1mulgsum 42977 |
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