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Mirrors > Home > MPE Home > Th. List > issubm2 | Structured version Visualization version GIF version |
Description: Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
issubm2.b | β’ π΅ = (Baseβπ) |
issubm2.z | β’ 0 = (0gβπ) |
issubm2.h | β’ π» = (π βΎs π) |
Ref | Expression |
---|---|
issubm2 | β’ (π β Mnd β (π β (SubMndβπ) β (π β π΅ β§ 0 β π β§ π» β Mnd))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubm2.b | . . 3 β’ π΅ = (Baseβπ) | |
2 | issubm2.z | . . 3 β’ 0 = (0gβπ) | |
3 | eqid 2726 | . . 3 β’ (+gβπ) = (+gβπ) | |
4 | 1, 2, 3 | issubm 18726 | . 2 β’ (π β Mnd β (π β (SubMndβπ) β (π β π΅ β§ 0 β π β§ βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π))) |
5 | issubm2.h | . . . . . . 7 β’ π» = (π βΎs π) | |
6 | 1, 3, 2, 5 | issubmnd 18692 | . . . . . 6 β’ ((π β Mnd β§ π β π΅ β§ 0 β π) β (π» β Mnd β βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π)) |
7 | 6 | bicomd 222 | . . . . 5 β’ ((π β Mnd β§ π β π΅ β§ 0 β π) β (βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π β π» β Mnd)) |
8 | 7 | 3expb 1117 | . . . 4 β’ ((π β Mnd β§ (π β π΅ β§ 0 β π)) β (βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π β π» β Mnd)) |
9 | 8 | pm5.32da 578 | . . 3 β’ (π β Mnd β (((π β π΅ β§ 0 β π) β§ βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π) β ((π β π΅ β§ 0 β π) β§ π» β Mnd))) |
10 | df-3an 1086 | . . 3 β’ ((π β π΅ β§ 0 β π β§ βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π) β ((π β π΅ β§ 0 β π) β§ βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π)) | |
11 | df-3an 1086 | . . 3 β’ ((π β π΅ β§ 0 β π β§ π» β Mnd) β ((π β π΅ β§ 0 β π) β§ π» β Mnd)) | |
12 | 9, 10, 11 | 3bitr4g 314 | . 2 β’ (π β Mnd β ((π β π΅ β§ 0 β π β§ βπ₯ β π βπ¦ β π (π₯(+gβπ)π¦) β π) β (π β π΅ β§ 0 β π β§ π» β Mnd))) |
13 | 4, 12 | bitrd 279 | 1 β’ (π β Mnd β (π β (SubMndβπ) β (π β π΅ β§ 0 β π β§ π» β Mnd))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 β wss 3943 βcfv 6536 (class class class)co 7404 Basecbs 17151 βΎs cress 17180 +gcplusg 17204 0gc0g 17392 Mndcmnd 18665 SubMndcsubmnd 18710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 |
This theorem is referenced by: issubmndb 18728 submss 18732 submid 18733 subm0cl 18734 submmnd 18736 subsubm 18739 idresefmnd 18822 cycsubmcmn 19807 unitsubm 20286 subrgsubm 20485 primrootscoprmpow 41477 |
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