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Mirrors > Home > MPE Home > Th. List > mulgnn0cld | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 18850. (Contributed by SN, 1-Feb-2025.) |
Ref | Expression |
---|---|
mulgnn0cld.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnn0cld.t | ⊢ · = (.g‘𝐺) |
mulgnn0cld.m | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
mulgnn0cld.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
mulgnn0cld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mulgnn0cld | ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnn0cld.m | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
2 | mulgnn0cld.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | mulgnn0cld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
4 | mulgnn0cld.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
5 | mulgnn0cld.t | . . 3 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | mulgnn0cl 18850 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1371 | 1 ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 ℕ0cn0 12371 Basecbs 17042 Mndcmnd 18515 .gcmg 18830 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-seq 13861 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mulg 18831 |
This theorem is referenced by: mulgnn0dir 18864 mhmmulg 18875 pwsmulg 18879 odmodnn0 19280 finodsubmsubg 19307 srgmulgass 19901 srgpcomp 19902 srgpcompp 19903 srgpcomppsc 19904 srgbinomlem1 19910 srgbinomlem2 19911 srgbinomlem4 19913 srgbinomlem 19914 pwsexpg 19996 lmodvsmmulgdi 20309 assamulgscmlem2 21255 mplcoe5lem 21391 mplcoe5 21392 evlslem3 21441 ply1moncl 21593 coe1pwmul 21601 ply1coefsupp 21617 ply1coe 21618 gsummoncoe1 21626 lply1binomsc 21629 evl1expd 21662 evl1scvarpw 21680 evl1scvarpwval 21681 evl1gsummon 21682 pmatcollpwscmatlem1 22089 mply1topmatcllem 22103 mply1topmatcl 22105 pm2mpghm 22116 monmat2matmon 22124 pm2mp 22125 chpscmatgsumbin 22144 chpscmatgsummon 22145 chfacfscmulcl 22157 chfacfscmul0 22158 chfacfpmmulcl 22161 chfacfpmmul0 22162 cpmadugsumlemB 22174 cpmadugsumlemC 22175 cpmadugsumlemF 22176 cayhamlem2 22184 cayhamlem4 22188 deg1pw 25436 plypf1 25524 lgsqrlem2 26646 lgsqrlem3 26647 lgsqrlem4 26648 omndmul2 31744 omndmul3 31745 omndmul 31746 isarchi2 31845 freshmansdream 31890 frobrhm 31891 evls1fpws 32088 ply1chr 32095 evlsbagval 40663 evlsexpval 40664 mhphflem 40672 mhphf 40673 hbtlem4 41355 lmodvsmdi 46352 ply1mulgsumlem4 46364 ply1mulgsum 46365 |
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