Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > subsubm | Structured version Visualization version Unicode version |
Description: A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
subsubm.h | ↾s |
Ref | Expression |
---|---|
subsubm | SubMnd SubMnd SubMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . . 8 | |
2 | 1 | submss 17350 | . . . . . . 7 SubMnd |
3 | 2 | adantl 482 | . . . . . 6 SubMnd SubMnd |
4 | subsubm.h | . . . . . . . 8 ↾s | |
5 | 4 | submbas 17355 | . . . . . . 7 SubMnd |
6 | 5 | adantr 481 | . . . . . 6 SubMnd SubMnd |
7 | 3, 6 | sseqtr4d 3642 | . . . . 5 SubMnd SubMnd |
8 | eqid 2622 | . . . . . . 7 | |
9 | 8 | submss 17350 | . . . . . 6 SubMnd |
10 | 9 | adantr 481 | . . . . 5 SubMnd SubMnd |
11 | 7, 10 | sstrd 3613 | . . . 4 SubMnd SubMnd |
12 | eqid 2622 | . . . . . . 7 | |
13 | 4, 12 | subm0 17356 | . . . . . 6 SubMnd |
14 | 13 | adantr 481 | . . . . 5 SubMnd SubMnd |
15 | eqid 2622 | . . . . . . 7 | |
16 | 15 | subm0cl 17352 | . . . . . 6 SubMnd |
17 | 16 | adantl 482 | . . . . 5 SubMnd SubMnd |
18 | 14, 17 | eqeltrd 2701 | . . . 4 SubMnd SubMnd |
19 | 4 | oveq1i 6660 | . . . . . . 7 ↾s ↾s ↾s |
20 | ressabs 15939 | . . . . . . 7 SubMnd ↾s ↾s ↾s | |
21 | 19, 20 | syl5eq 2668 | . . . . . 6 SubMnd ↾s ↾s |
22 | 7, 21 | syldan 487 | . . . . 5 SubMnd SubMnd ↾s ↾s |
23 | eqid 2622 | . . . . . . 7 ↾s ↾s | |
24 | 23 | submmnd 17354 | . . . . . 6 SubMnd ↾s |
25 | 24 | adantl 482 | . . . . 5 SubMnd SubMnd ↾s |
26 | 22, 25 | eqeltrrd 2702 | . . . 4 SubMnd SubMnd ↾s |
27 | submrcl 17346 | . . . . . 6 SubMnd | |
28 | 27 | adantr 481 | . . . . 5 SubMnd SubMnd |
29 | eqid 2622 | . . . . . 6 ↾s ↾s | |
30 | 8, 12, 29 | issubm2 17348 | . . . . 5 SubMnd ↾s |
31 | 28, 30 | syl 17 | . . . 4 SubMnd SubMnd SubMnd ↾s |
32 | 11, 18, 26, 31 | mpbir3and 1245 | . . 3 SubMnd SubMnd SubMnd |
33 | 32, 7 | jca 554 | . 2 SubMnd SubMnd SubMnd |
34 | simprr 796 | . . . 4 SubMnd SubMnd | |
35 | 5 | adantr 481 | . . . 4 SubMnd SubMnd |
36 | 34, 35 | sseqtrd 3641 | . . 3 SubMnd SubMnd |
37 | 13 | adantr 481 | . . . 4 SubMnd SubMnd |
38 | 12 | subm0cl 17352 | . . . . 5 SubMnd |
39 | 38 | ad2antrl 764 | . . . 4 SubMnd SubMnd |
40 | 37, 39 | eqeltrrd 2702 | . . 3 SubMnd SubMnd |
41 | 21 | adantrl 752 | . . . 4 SubMnd SubMnd ↾s ↾s |
42 | 29 | submmnd 17354 | . . . . 5 SubMnd ↾s |
43 | 42 | ad2antrl 764 | . . . 4 SubMnd SubMnd ↾s |
44 | 41, 43 | eqeltrd 2701 | . . 3 SubMnd SubMnd ↾s |
45 | 4 | submmnd 17354 | . . . . 5 SubMnd |
46 | 45 | adantr 481 | . . . 4 SubMnd SubMnd |
47 | 1, 15, 23 | issubm2 17348 | . . . 4 SubMnd ↾s |
48 | 46, 47 | syl 17 | . . 3 SubMnd SubMnd SubMnd ↾s |
49 | 36, 40, 44, 48 | mpbir3and 1245 | . 2 SubMnd SubMnd SubMnd |
50 | 33, 49 | impbida 877 | 1 SubMnd SubMnd SubMnd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 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 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 df-submnd 17336 |
This theorem is referenced by: zrhpsgnmhm 19930 amgmlem 24716 nn0archi 29843 amgmwlem 42548 amgmlemALT 42549 |
Copyright terms: Public domain | W3C validator |