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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 2821 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mnd) | |
5 | ssidd 3989 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐵 ⊆ 𝐵) | |
6 | 1, 3 | mndcl 17909 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
7 | eqid 2821 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 1, 7 | mndidcl 17916 | . 2 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 18181 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 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 ℕ0cn0 11886 Basecbs 16473 +gcplusg 16555 0gc0g 16703 Mndcmnd 17901 .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-mulg 18165 |
This theorem is referenced by: mulgnn0dir 18197 mulgnn0ass 18203 mhmmulg 18208 pwsmulg 18212 cycsubm 18285 odmodnn0 18599 mulgmhm 18879 srgmulgass 19212 srgpcomp 19213 srgpcompp 19214 srgpcomppsc 19215 srgbinomlem1 19221 srgbinomlem2 19222 srgbinomlem4 19224 srgbinomlem 19225 lmodvsmmulgdi 19600 assamulgscmlem2 20059 mplcoe5lem 20178 mplcoe5 20179 psrbagev1 20220 evlslem3 20223 ply1moncl 20369 coe1pwmul 20377 ply1coefsupp 20393 ply1coe 20394 gsummoncoe1 20402 lply1binomsc 20405 evl1expd 20438 evl1scvarpw 20456 evl1scvarpwval 20457 evl1gsummon 20458 pmatcollpwscmatlem1 21327 mply1topmatcllem 21341 mply1topmatcl 21343 pm2mpghm 21354 monmat2matmon 21362 pm2mp 21363 chpscmatgsumbin 21382 chpscmatgsummon 21383 chfacfscmulcl 21395 chfacfscmul0 21396 chfacfpmmulcl 21399 chfacfpmmul0 21400 cpmadugsumlemB 21412 cpmadugsumlemC 21413 cpmadugsumlemF 21414 cayhamlem2 21422 cayhamlem4 21426 deg1pw 24643 plypf1 24731 lgsqrlem2 25851 lgsqrlem3 25852 lgsqrlem4 25853 omndmul2 30641 omndmul3 30642 omndmul 30643 isarchi2 30742 freshmansdream 30787 hbtlem4 39606 lmodvsmdi 44328 ply1mulgsumlem4 44341 ply1mulgsum 44342 |
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