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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 2651 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mnd) | |
5 | ssid 3657 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐵 ⊆ 𝐵) |
7 | 1, 3 | mndcl 17348 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
8 | eqid 2651 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 1, 8 | mndidcl 17355 | . 2 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
10 | 1, 2, 3, 4, 6, 7, 8, 9 | mulgnn0subcl 17601 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ‘cfv 5926 (class class class)co 6690 ℕ0cn0 11330 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Mndcmnd 17341 .gcmg 17587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-seq 12842 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mulg 17588 |
This theorem is referenced by: mulgnn0dir 17618 mulgnn0ass 17625 mhmmulg 17630 pwsmulg 17634 odmodnn0 18005 mulgmhm 18279 srgmulgass 18577 srgpcomp 18578 srgpcompp 18579 srgpcomppsc 18580 srgbinomlem1 18586 srgbinomlem2 18587 srgbinomlem4 18589 srgbinomlem 18590 lmodvsmmulgdi 18946 assamulgscmlem2 19397 mplcoe5lem 19515 mplcoe5 19516 psrbagev1 19558 evlslem3 19562 ply1moncl 19689 coe1pwmul 19697 ply1coefsupp 19713 ply1coe 19714 gsummoncoe1 19722 lply1binomsc 19725 evl1expd 19757 evl1scvarpw 19775 evl1scvarpwval 19776 evl1gsummon 19777 pmatcollpwscmatlem1 20642 mply1topmatcllem 20656 mply1topmatcl 20658 pm2mpghm 20669 monmat2matmon 20677 pm2mp 20678 chpscmatgsumbin 20697 chpscmatgsummon 20698 chfacfscmulcl 20710 chfacfscmul0 20711 chfacfpmmulcl 20714 chfacfpmmul0 20715 cpmadugsumlemB 20727 cpmadugsumlemC 20728 cpmadugsumlemF 20729 cayhamlem2 20737 cayhamlem4 20741 deg1pw 23925 plypf1 24013 lgsqrlem2 25117 lgsqrlem3 25118 lgsqrlem4 25119 omndmul2 29840 omndmul3 29841 omndmul 29842 isarchi2 29867 hbtlem4 38013 lmodvsmdi 42488 ply1mulgsumlem4 42502 ply1mulgsum 42503 |
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