Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mnd4g | Structured version Visualization version Unicode version |
Description: Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
mndcl.b | |
mndcl.p | |
mnd4g.1 | |
mnd4g.2 | |
mnd4g.3 | |
mnd4g.4 | |
mnd4g.5 | |
mnd4g.6 |
Ref | Expression |
---|---|
mnd4g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndcl.b | . . . 4 | |
2 | mndcl.p | . . . 4 | |
3 | mnd4g.1 | . . . 4 | |
4 | mnd4g.3 | . . . 4 | |
5 | mnd4g.4 | . . . 4 | |
6 | mnd4g.5 | . . . 4 | |
7 | mnd4g.6 | . . . 4 | |
8 | 1, 2, 3, 4, 5, 6, 7 | mnd12g 17306 | . . 3 |
9 | 8 | oveq2d 6666 | . 2 |
10 | mnd4g.2 | . . 3 | |
11 | 1, 2 | mndcl 17301 | . . . 4 |
12 | 3, 5, 6, 11 | syl3anc 1326 | . . 3 |
13 | 1, 2 | mndass 17302 | . . 3 |
14 | 3, 10, 4, 12, 13 | syl13anc 1328 | . 2 |
15 | 1, 2 | mndcl 17301 | . . . 4 |
16 | 3, 4, 6, 15 | syl3anc 1326 | . . 3 |
17 | 1, 2 | mndass 17302 | . . 3 |
18 | 3, 10, 5, 16, 17 | syl13anc 1328 | . 2 |
19 | 9, 14, 18 | 3eqtr4d 2666 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmnd 17294 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 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-iota 5851 df-fv 5896 df-ov 6653 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: lsmsubm 18068 pj1ghm 18116 cmn4 18212 gsumzaddlem 18321 |
Copyright terms: Public domain | W3C validator |