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Mirrors > Home > MPE Home > Th. List > 0mhm | Structured version Visualization version Unicode version |
Description: The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
0mhm.z | |
0mhm.b |
Ref | Expression |
---|---|
0mhm | MndHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 | |
2 | eqid 2622 | . . . . . 6 | |
3 | 0mhm.z | . . . . . 6 | |
4 | 2, 3 | mndidcl 17308 | . . . . 5 |
5 | 4 | adantl 482 | . . . 4 |
6 | fconst6g 6094 | . . . 4 | |
7 | 5, 6 | syl 17 | . . 3 |
8 | simpr 477 | . . . . . . 7 | |
9 | eqid 2622 | . . . . . . . . 9 | |
10 | 2, 9, 3 | mndlid 17311 | . . . . . . . 8 |
11 | 10 | eqcomd 2628 | . . . . . . 7 |
12 | 8, 5, 11 | syl2anc 693 | . . . . . 6 |
13 | 12 | adantr 481 | . . . . 5 |
14 | 0mhm.b | . . . . . . . . 9 | |
15 | eqid 2622 | . . . . . . . . 9 | |
16 | 14, 15 | mndcl 17301 | . . . . . . . 8 |
17 | 16 | 3expb 1266 | . . . . . . 7 |
18 | 17 | adantlr 751 | . . . . . 6 |
19 | fvex 6201 | . . . . . . . 8 | |
20 | 3, 19 | eqeltri 2697 | . . . . . . 7 |
21 | 20 | fvconst2 6469 | . . . . . 6 |
22 | 18, 21 | syl 17 | . . . . 5 |
23 | 20 | fvconst2 6469 | . . . . . . 7 |
24 | 20 | fvconst2 6469 | . . . . . . 7 |
25 | 23, 24 | oveqan12d 6669 | . . . . . 6 |
26 | 25 | adantl 482 | . . . . 5 |
27 | 13, 22, 26 | 3eqtr4d 2666 | . . . 4 |
28 | 27 | ralrimivva 2971 | . . 3 |
29 | eqid 2622 | . . . . . 6 | |
30 | 14, 29 | mndidcl 17308 | . . . . 5 |
31 | 30 | adantr 481 | . . . 4 |
32 | 20 | fvconst2 6469 | . . . 4 |
33 | 31, 32 | syl 17 | . . 3 |
34 | 7, 28, 33 | 3jca 1242 | . 2 |
35 | 14, 2, 15, 9, 29, 3 | ismhm 17337 | . 2 MndHom |
36 | 1, 34, 35 | sylanbrc 698 | 1 MndHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cvv 3200 csn 4177 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 MndHom cmhm 17333 |
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 |
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-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 |
This theorem is referenced by: 0ghm 17674 |
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