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Mirrors > Home > MPE Home > Th. List > ghmcmn | Structured version Visualization version Unicode version |
Description: The image of a commutative monoid under a group homomorphism is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
Ref | Expression |
---|---|
ghmabl.x | |
ghmabl.y | |
ghmabl.p | |
ghmabl.q | |
ghmabl.f | |
ghmabl.1 | |
ghmcmn.3 | CMnd |
Ref | Expression |
---|---|
ghmcmn | CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmabl.f | . . 3 | |
2 | ghmabl.x | . . 3 | |
3 | ghmabl.y | . . 3 | |
4 | ghmabl.p | . . 3 | |
5 | ghmabl.q | . . 3 | |
6 | ghmabl.1 | . . 3 | |
7 | ghmcmn.3 | . . . 4 CMnd | |
8 | cmnmnd 18208 | . . . 4 CMnd | |
9 | 7, 8 | syl 17 | . . 3 |
10 | 1, 2, 3, 4, 5, 6, 9 | mhmmnd 17537 | . 2 |
11 | simp-6l 810 | . . . . . . . . . . 11 | |
12 | 11, 7 | syl 17 | . . . . . . . . . 10 CMnd |
13 | simp-4r 807 | . . . . . . . . . 10 | |
14 | simplr 792 | . . . . . . . . . 10 | |
15 | 2, 4 | cmncom 18209 | . . . . . . . . . 10 CMnd |
16 | 12, 13, 14, 15 | syl3anc 1326 | . . . . . . . . 9 |
17 | 16 | fveq2d 6195 | . . . . . . . 8 |
18 | 11, 1 | syl3an1 1359 | . . . . . . . . 9 |
19 | 18, 13, 14 | mhmlem 17535 | . . . . . . . 8 |
20 | 18, 14, 13 | mhmlem 17535 | . . . . . . . 8 |
21 | 17, 19, 20 | 3eqtr3d 2664 | . . . . . . 7 |
22 | simpllr 799 | . . . . . . . 8 | |
23 | simpr 477 | . . . . . . . 8 | |
24 | 22, 23 | oveq12d 6668 | . . . . . . 7 |
25 | 23, 22 | oveq12d 6668 | . . . . . . 7 |
26 | 21, 24, 25 | 3eqtr3d 2664 | . . . . . 6 |
27 | foelrni 6244 | . . . . . . . . 9 | |
28 | 6, 27 | sylan 488 | . . . . . . . 8 |
29 | 28 | adantlr 751 | . . . . . . 7 |
30 | 29 | ad2antrr 762 | . . . . . 6 |
31 | 26, 30 | r19.29a 3078 | . . . . 5 |
32 | foelrni 6244 | . . . . . . 7 | |
33 | 6, 32 | sylan 488 | . . . . . 6 |
34 | 33 | adantr 481 | . . . . 5 |
35 | 31, 34 | r19.29a 3078 | . . . 4 |
36 | 35 | anasss 679 | . . 3 |
37 | 36 | ralrimivva 2971 | . 2 |
38 | 3, 5 | iscmn 18200 | . 2 CMnd |
39 | 10, 37, 38 | sylanbrc 698 | 1 CMnd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wfo 5886 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmnd 17294 CMndccmn 18193 |
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-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-fo 5894 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cmn 18195 |
This theorem is referenced by: ghmabl 18238 |
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