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Mirrors > Home > MPE Home > Th. List > submcl | Structured version Visualization version GIF version |
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
submcl.p | β’ + = (+gβπ) |
Ref | Expression |
---|---|
submcl | β’ ((π β (SubMndβπ) β§ π β π β§ π β π) β (π + π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submrcl 18725 | . . . . . . 7 β’ (π β (SubMndβπ) β π β Mnd) | |
2 | eqid 2726 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
3 | eqid 2726 | . . . . . . . 8 β’ (0gβπ) = (0gβπ) | |
4 | submcl.p | . . . . . . . 8 β’ + = (+gβπ) | |
5 | 2, 3, 4 | issubm 18726 | . . . . . . 7 β’ (π β Mnd β (π β (SubMndβπ) β (π β (Baseβπ) β§ (0gβπ) β π β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π))) |
6 | 1, 5 | syl 17 | . . . . . 6 β’ (π β (SubMndβπ) β (π β (SubMndβπ) β (π β (Baseβπ) β§ (0gβπ) β π β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π))) |
7 | 6 | ibi 267 | . . . . 5 β’ (π β (SubMndβπ) β (π β (Baseβπ) β§ (0gβπ) β π β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π)) |
8 | 7 | simp3d 1141 | . . . 4 β’ (π β (SubMndβπ) β βπ₯ β π βπ¦ β π (π₯ + π¦) β π) |
9 | ovrspc2v 7430 | . . . 4 β’ (((π β π β§ π β π) β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π) β (π + π) β π) | |
10 | 8, 9 | sylan2 592 | . . 3 β’ (((π β π β§ π β π) β§ π β (SubMndβπ)) β (π + π) β π) |
11 | 10 | ancoms 458 | . 2 β’ ((π β (SubMndβπ) β§ (π β π β§ π β π)) β (π + π) β π) |
12 | 11 | 3impb 1112 | 1 β’ ((π β (SubMndβπ) β§ π β π β§ π β π) β (π + π) β π) |
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 +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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-submnd 18712 |
This theorem is referenced by: resmhm 18743 mhmima 18748 gsumwsubmcl 18760 submmulgcl 19042 symggen 19388 lsmsubm 19571 smndlsmidm 19574 gsumzadd 19840 gsumzoppg 19862 |
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