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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 2797 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | id 22 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mnd) | |
5 | ssidd 3917 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐵 ⊆ 𝐵) | |
6 | 1, 3 | mndcl 17744 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
7 | eqid 2797 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 1, 7 | mndidcl 17751 | . 2 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 18000 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 (class class class)co 7023 ℕ0cn0 11751 Basecbs 16316 +gcplusg 16398 0gc0g 16546 Mndcmnd 17737 .gcmg 17985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-seq 13224 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mulg 17986 |
This theorem is referenced by: mulgnn0dir 18015 mulgnn0ass 18021 mhmmulg 18026 pwsmulg 18030 odmodnn0 18403 mulgmhm 18677 srgmulgass 18975 srgpcomp 18976 srgpcompp 18977 srgpcomppsc 18978 srgbinomlem1 18984 srgbinomlem2 18985 srgbinomlem4 18987 srgbinomlem 18988 lmodvsmmulgdi 19363 assamulgscmlem2 19821 mplcoe5lem 19939 mplcoe5 19940 psrbagev1 19981 evlslem3 19984 ply1moncl 20126 coe1pwmul 20134 ply1coefsupp 20150 ply1coe 20151 gsummoncoe1 20159 lply1binomsc 20162 evl1expd 20194 evl1scvarpw 20212 evl1scvarpwval 20213 evl1gsummon 20214 pmatcollpwscmatlem1 21085 mply1topmatcllem 21099 mply1topmatcl 21101 pm2mpghm 21112 monmat2matmon 21120 pm2mp 21121 chpscmatgsumbin 21140 chpscmatgsummon 21141 chfacfscmulcl 21153 chfacfscmul0 21154 chfacfpmmulcl 21157 chfacfpmmul0 21158 cpmadugsumlemB 21170 cpmadugsumlemC 21171 cpmadugsumlemF 21172 cayhamlem2 21180 cayhamlem4 21184 deg1pw 24401 plypf1 24489 lgsqrlem2 25609 lgsqrlem3 25610 lgsqrlem4 25611 omndmul2 30369 omndmul3 30370 omndmul 30371 isarchi2 30448 freshmansdream 30509 hbtlem4 39232 lmodvsmdi 43932 ply1mulgsumlem4 43945 ply1mulgsum 43946 |
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