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Mirrors > Home > MPE Home > Th. List > submcl | Structured version Visualization version Unicode version |
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
submcl.p |
Ref | Expression |
---|---|
submcl | SubMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submrcl 17346 | . . . . . . 7 SubMnd | |
2 | eqid 2622 | . . . . . . . 8 | |
3 | eqid 2622 | . . . . . . . 8 | |
4 | submcl.p | . . . . . . . 8 | |
5 | 2, 3, 4 | issubm 17347 | . . . . . . 7 SubMnd |
6 | 1, 5 | syl 17 | . . . . . 6 SubMnd SubMnd |
7 | 6 | ibi 256 | . . . . 5 SubMnd |
8 | 7 | simp3d 1075 | . . . 4 SubMnd |
9 | ovrspc2v 6672 | . . . 4 | |
10 | 8, 9 | sylan2 491 | . . 3 SubMnd |
11 | 10 | ancoms 469 | . 2 SubMnd |
12 | 11 | 3impb 1260 | 1 SubMnd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 SubMndcsubmnd 17334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-submnd 17336 |
This theorem is referenced by: resmhm 17359 mhmima 17363 gsumwsubmcl 17375 submmulgcl 17585 symggen 17890 lsmsubm 18068 gsumzadd 18322 gsumzoppg 18344 |
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