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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 | โข ๐ต = (Baseโ๐บ) |
mulgnncl.t | โข ยท = (.gโ๐บ) |
Ref | Expression |
---|---|
mulgnn0cl | โข ((๐บ โ Mnd โง ๐ โ โ0 โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnncl.b | . 2 โข ๐ต = (Baseโ๐บ) | |
2 | mulgnncl.t | . 2 โข ยท = (.gโ๐บ) | |
3 | eqid 2732 | . 2 โข (+gโ๐บ) = (+gโ๐บ) | |
4 | id 22 | . 2 โข (๐บ โ Mnd โ ๐บ โ Mnd) | |
5 | ssidd 4005 | . 2 โข (๐บ โ Mnd โ ๐ต โ ๐ต) | |
6 | 1, 3 | mndcl 18635 | . 2 โข ((๐บ โ Mnd โง ๐ฅ โ ๐ต โง ๐ฆ โ ๐ต) โ (๐ฅ(+gโ๐บ)๐ฆ) โ ๐ต) |
7 | eqid 2732 | . 2 โข (0gโ๐บ) = (0gโ๐บ) | |
8 | 1, 7 | mndidcl 18642 | . 2 โข (๐บ โ Mnd โ (0gโ๐บ) โ ๐ต) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 18969 | 1 โข ((๐บ โ Mnd โง ๐ โ โ0 โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1087 = wceq 1541 โ wcel 2106 โcfv 6543 (class class class)co 7411 โ0cn0 12474 Basecbs 17146 +gcplusg 17199 0gc0g 17387 Mndcmnd 18627 .gcmg 18952 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-seq 13969 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mulg 18953 |
This theorem is referenced by: mulgnn0cld 18977 mulgnn0ass 18992 cycsubm 19081 mulgmhm 19697 psrbagev1 21644 psrbagev1OLD 21645 |
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