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Mirrors > Home > MPE Home > Th. List > Mathboxes > c0mgm | Structured version Visualization version Unicode version |
Description: The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
c0mhm.b | |
c0mhm.0 | |
c0mhm.h |
Ref | Expression |
---|---|
c0mgm | Mgm MgmHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndmgm 17300 | . . 3 Mgm | |
2 | 1 | anim2i 593 | . 2 Mgm Mgm Mgm |
3 | eqid 2622 | . . . . . . 7 | |
4 | c0mhm.0 | . . . . . . 7 | |
5 | 3, 4 | mndidcl 17308 | . . . . . 6 |
6 | 5 | adantl 482 | . . . . 5 Mgm |
7 | 6 | adantr 481 | . . . 4 Mgm |
8 | c0mhm.h | . . . 4 | |
9 | 7, 8 | fmptd 6385 | . . 3 Mgm |
10 | 5 | ancli 574 | . . . . . . . 8 |
11 | 10 | adantl 482 | . . . . . . 7 Mgm |
12 | eqid 2622 | . . . . . . . 8 | |
13 | 3, 12, 4 | mndlid 17311 | . . . . . . 7 |
14 | 11, 13 | syl 17 | . . . . . 6 Mgm |
15 | 14 | adantr 481 | . . . . 5 Mgm |
16 | 8 | a1i 11 | . . . . . . 7 Mgm |
17 | eqidd 2623 | . . . . . . 7 Mgm | |
18 | simprl 794 | . . . . . . 7 Mgm | |
19 | 6 | adantr 481 | . . . . . . 7 Mgm |
20 | 16, 17, 18, 19 | fvmptd 6288 | . . . . . 6 Mgm |
21 | eqidd 2623 | . . . . . . 7 Mgm | |
22 | simprr 796 | . . . . . . 7 Mgm | |
23 | 16, 21, 22, 19 | fvmptd 6288 | . . . . . 6 Mgm |
24 | 20, 23 | oveq12d 6668 | . . . . 5 Mgm |
25 | eqidd 2623 | . . . . . 6 Mgm | |
26 | c0mhm.b | . . . . . . . . 9 | |
27 | eqid 2622 | . . . . . . . . 9 | |
28 | 26, 27 | mgmcl 17245 | . . . . . . . 8 Mgm |
29 | 28 | 3expb 1266 | . . . . . . 7 Mgm |
30 | 29 | adantlr 751 | . . . . . 6 Mgm |
31 | 16, 25, 30, 19 | fvmptd 6288 | . . . . 5 Mgm |
32 | 15, 24, 31 | 3eqtr4rd 2667 | . . . 4 Mgm |
33 | 32 | ralrimivva 2971 | . . 3 Mgm |
34 | 9, 33 | jca 554 | . 2 Mgm |
35 | 26, 3, 27, 12 | ismgmhm 41783 | . 2 MgmHom Mgm Mgm |
36 | 2, 34, 35 | sylanbrc 698 | 1 Mgm MgmHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cmpt 4729 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 Mgmcmgm 17240 cmnd 17294 MgmHom cmgmhm 41777 |
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-csb 3534 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-res 5126 df-ima 5127 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-mgmhm 41779 |
This theorem is referenced by: c0rnghm 41913 |
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