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Mirrors > Home > MPE Home > Th. List > issubmd | Structured version Visualization version Unicode version |
Description: Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
issubmd.b | |
issubmd.p | |
issubmd.z | |
issubmd.m | |
issubmd.cz | |
issubmd.cp | |
issubmd.ch | |
issubmd.th | |
issubmd.ta | |
issubmd.et |
Ref | Expression |
---|---|
issubmd | SubMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3687 | . . 3 | |
2 | 1 | a1i 11 | . 2 |
3 | issubmd.m | . . . 4 | |
4 | issubmd.b | . . . . 5 | |
5 | issubmd.z | . . . . 5 | |
6 | 4, 5 | mndidcl 17308 | . . . 4 |
7 | 3, 6 | syl 17 | . . 3 |
8 | issubmd.cz | . . 3 | |
9 | issubmd.ch | . . . 4 | |
10 | 9 | elrab 3363 | . . 3 |
11 | 7, 8, 10 | sylanbrc 698 | . 2 |
12 | issubmd.th | . . . . . 6 | |
13 | 12 | elrab 3363 | . . . . 5 |
14 | issubmd.ta | . . . . . 6 | |
15 | 14 | elrab 3363 | . . . . 5 |
16 | 13, 15 | anbi12i 733 | . . . 4 |
17 | 3 | adantr 481 | . . . . . 6 |
18 | simprll 802 | . . . . . 6 | |
19 | simprrl 804 | . . . . . 6 | |
20 | issubmd.p | . . . . . . 7 | |
21 | 4, 20 | mndcl 17301 | . . . . . 6 |
22 | 17, 18, 19, 21 | syl3anc 1326 | . . . . 5 |
23 | an4 865 | . . . . . 6 | |
24 | issubmd.cp | . . . . . 6 | |
25 | 23, 24 | sylan2b 492 | . . . . 5 |
26 | issubmd.et | . . . . . 6 | |
27 | 26 | elrab 3363 | . . . . 5 |
28 | 22, 25, 27 | sylanbrc 698 | . . . 4 |
29 | 16, 28 | sylan2b 492 | . . 3 |
30 | 29 | ralrimivva 2971 | . 2 |
31 | 4, 5, 20 | issubm 17347 | . . 3 SubMnd |
32 | 3, 31 | syl 17 | . 2 SubMnd |
33 | 2, 11, 30, 32 | mpbir3and 1245 | 1 SubMnd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 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-reu 2919 df-rmo 2920 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-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 |
This theorem is referenced by: mrcmndind 17366 |
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