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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 18754 | . . . . . . 7 β’ (π β (SubMndβπ) β π β Mnd) | |
2 | eqid 2728 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
3 | eqid 2728 | . . . . . . . 8 β’ (0gβπ) = (0gβπ) | |
4 | submcl.p | . . . . . . . 8 β’ + = (+gβπ) | |
5 | 2, 3, 4 | issubm 18755 | . . . . . . 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 1142 | . . . 4 β’ (π β (SubMndβπ) β βπ₯ β π βπ¦ β π (π₯ + π¦) β π) |
9 | ovrspc2v 7446 | . . . 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 1113 | 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 1085 = wceq 1534 β wcel 2099 βwral 3058 β wss 3947 βcfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 0gc0g 17421 Mndcmnd 18694 SubMndcsubmnd 18739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-submnd 18741 |
This theorem is referenced by: resmhm 18772 mhmima 18777 gsumwsubmcl 18789 submmulgcl 19072 symggen 19425 lsmsubm 19608 smndlsmidm 19611 gsumzadd 19877 gsumzoppg 19899 submcld 32768 erler 32992 rlocaddval 32995 rlocmulval 32996 |
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