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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 2727 | . . 3 β’ (+gβπ) = (+gβπ) | |
4 | 1, 2, 3 | issubm 18760 | . 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 18726 | . . . . . 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 577 | . . 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 313 | . 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 278 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3057 β wss 3947 βcfv 6551 (class class class)co 7424 Basecbs 17185 βΎs cress 17214 +gcplusg 17238 0gc0g 17426 Mndcmnd 18699 SubMndcsubmnd 18744 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 |
This theorem is referenced by: issubmndb 18762 submss 18766 submid 18767 subm0cl 18768 submmnd 18770 subsubm 18773 idresefmnd 18856 cycsubmcmn 19849 unitsubm 20330 subrgsubm 20529 primrootscoprmpow 41574 |
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