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Mirrors > Home > MPE Home > Th. List > submmulgcl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
submmulgcl.t | ⊢ ∙ = (.g‘𝐺) |
Ref | Expression |
---|---|
submmulgcl | ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | submmulgcl.t | . 2 ⊢ ∙ = (.g‘𝐺) | |
3 | eqid 2824 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | submrcl 17698 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd) | |
5 | 1 | submss 17702 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
6 | 3 | submcl 17705 | . 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) |
7 | eqid 2824 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 7 | subm0cl 17704 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑆) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 17907 | 1 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 (class class class)co 6904 ℕ0cn0 11617 Basecbs 16221 +gcplusg 16304 0gc0g 16452 Mndcmnd 17646 SubMndcsubmnd 17686 .gcmg 17893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-seq 13095 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 |
This theorem is referenced by: cmodscexp 23289 plypf1 24366 |
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