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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 2733 | . 2 โข (+gโ๐บ) = (+gโ๐บ) | |
4 | id 22 | . 2 โข (๐บ โ Mnd โ ๐บ โ Mnd) | |
5 | ssidd 4006 | . 2 โข (๐บ โ Mnd โ ๐ต โ ๐ต) | |
6 | 1, 3 | mndcl 18633 | . 2 โข ((๐บ โ Mnd โง ๐ฅ โ ๐ต โง ๐ฆ โ ๐ต) โ (๐ฅ(+gโ๐บ)๐ฆ) โ ๐ต) |
7 | eqid 2733 | . 2 โข (0gโ๐บ) = (0gโ๐บ) | |
8 | 1, 7 | mndidcl 18640 | . 2 โข (๐บ โ Mnd โ (0gโ๐บ) โ ๐ต) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 18967 | 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 1088 = wceq 1542 โ wcel 2107 โcfv 6544 (class class class)co 7409 โ0cn0 12472 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Mndcmnd 18625 .gcmg 18950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-seq 13967 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mulg 18951 |
This theorem is referenced by: mulgnn0cld 18975 mulgnn0ass 18990 cycsubm 19079 mulgmhm 19695 psrbagev1 21638 psrbagev1OLD 21639 |
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