Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > issubmnd | Structured version Visualization version Unicode version |
Description: Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
issubmnd.b | |
issubmnd.p | |
issubmnd.z | |
issubmnd.h | ↾s |
Ref | Expression |
---|---|
issubmnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 792 | . . . . 5 | |
2 | simprl 794 | . . . . . 6 | |
3 | simpll2 1101 | . . . . . . 7 | |
4 | issubmnd.h | . . . . . . . 8 ↾s | |
5 | issubmnd.b | . . . . . . . 8 | |
6 | 4, 5 | ressbas2 15931 | . . . . . . 7 |
7 | 3, 6 | syl 17 | . . . . . 6 |
8 | 2, 7 | eleqtrd 2703 | . . . . 5 |
9 | simprr 796 | . . . . . 6 | |
10 | 9, 7 | eleqtrd 2703 | . . . . 5 |
11 | eqid 2622 | . . . . . 6 | |
12 | eqid 2622 | . . . . . 6 | |
13 | 11, 12 | mndcl 17301 | . . . . 5 |
14 | 1, 8, 10, 13 | syl3anc 1326 | . . . 4 |
15 | fvex 6201 | . . . . . . . . . 10 | |
16 | 5, 15 | eqeltri 2697 | . . . . . . . . 9 |
17 | 16 | ssex 4802 | . . . . . . . 8 |
18 | 17 | 3ad2ant2 1083 | . . . . . . 7 |
19 | issubmnd.p | . . . . . . . 8 | |
20 | 4, 19 | ressplusg 15993 | . . . . . . 7 |
21 | 18, 20 | syl 17 | . . . . . 6 |
22 | 21 | ad2antrr 762 | . . . . 5 |
23 | 22 | oveqd 6667 | . . . 4 |
24 | 14, 23, 7 | 3eltr4d 2716 | . . 3 |
25 | 24 | ralrimivva 2971 | . 2 |
26 | simpl2 1065 | . . . 4 | |
27 | 26, 6 | syl 17 | . . 3 |
28 | 21 | adantr 481 | . . 3 |
29 | ovrspc2v 6672 | . . . . . 6 | |
30 | 29 | ancoms 469 | . . . . 5 |
31 | 30 | 3impb 1260 | . . . 4 |
32 | 31 | 3adant1l 1318 | . . 3 |
33 | 26 | sseld 3602 | . . . . . 6 |
34 | 26 | sseld 3602 | . . . . . 6 |
35 | 26 | sseld 3602 | . . . . . 6 |
36 | 33, 34, 35 | 3anim123d 1406 | . . . . 5 |
37 | 36 | imp 445 | . . . 4 |
38 | simpl1 1064 | . . . . 5 | |
39 | 5, 19 | mndass 17302 | . . . . 5 |
40 | 38, 39 | sylan 488 | . . . 4 |
41 | 37, 40 | syldan 487 | . . 3 |
42 | simpl3 1066 | . . 3 | |
43 | 26 | sselda 3603 | . . . 4 |
44 | issubmnd.z | . . . . . 6 | |
45 | 5, 19, 44 | mndlid 17311 | . . . . 5 |
46 | 38, 45 | sylan 488 | . . . 4 |
47 | 43, 46 | syldan 487 | . . 3 |
48 | 5, 19, 44 | mndrid 17312 | . . . . 5 |
49 | 38, 48 | sylan 488 | . . . 4 |
50 | 43, 49 | syldan 487 | . . 3 |
51 | 27, 28, 32, 41, 42, 47, 50 | ismndd 17313 | . 2 |
52 | 25, 51 | impbida 877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cvv 3200 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cplusg 15941 c0g 16100 cmnd 17294 |
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 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: issubm2 17348 |
Copyright terms: Public domain | W3C validator |