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Mirrors > Home > ILE Home > Th. List > mndissubm | Unicode version |
Description: If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.) |
Ref | Expression |
---|---|
mndissubm.b | |
mndissubm.s | |
mndissubm.z |
Ref | Expression |
---|---|
mndissubm | SubMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1003 | . . 3 | |
2 | simpr2 1004 | . . 3 | |
3 | mndmgm 12688 | . . . . . . 7 Mgm | |
4 | mndmgm 12688 | . . . . . . 7 Mgm | |
5 | 3, 4 | anim12i 338 | . . . . . 6 Mgm Mgm |
6 | 5 | ad2antrr 488 | . . . . 5 Mgm Mgm |
7 | 3simpb 995 | . . . . . 6 | |
8 | 7 | ad2antlr 489 | . . . . 5 |
9 | simpr 110 | . . . . 5 | |
10 | mndissubm.b | . . . . . 6 | |
11 | mndissubm.s | . . . . . 6 | |
12 | 10, 11 | mgmsscl 12645 | . . . . 5 Mgm Mgm |
13 | 6, 8, 9, 12 | syl3anc 1238 | . . . 4 |
14 | 13 | ralrimivva 2557 | . . 3 |
15 | mndissubm.z | . . . . 5 | |
16 | eqid 2175 | . . . . 5 | |
17 | 10, 15, 16 | issubm 12725 | . . . 4 SubMnd |
18 | 17 | ad2antrr 488 | . . 3 SubMnd |
19 | 1, 2, 14, 18 | mpbir3and 1180 | . 2 SubMnd |
20 | 19 | ex 115 | 1 SubMnd |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 wral 2453 wss 3127 cxp 4618 cres 4622 cfv 5208 (class class class)co 5865 cbs 12428 cplusg 12492 c0g 12626 Mgmcmgm 12638 cmnd 12682 SubMndcsubmnd 12712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 df-ov 5868 df-inn 8891 df-2 8949 df-ndx 12431 df-slot 12432 df-base 12434 df-plusg 12505 df-mgm 12640 df-sgrp 12673 df-mnd 12683 df-submnd 12714 |
This theorem is referenced by: (None) |
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