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Mirrors > Home > MPE Home > Th. List > ismnd | Structured version Visualization version Unicode version |
Description: The predicate "is a monoid". This is the definig theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 17301), whose operation is associative (so, a semigroup, see also mndass 17302) and has a two-sided neutral element (see mndid 17303). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
Ref | Expression |
---|---|
ismnd.b | |
ismnd.p |
Ref | Expression |
---|---|
ismnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismnd.b | . . 3 | |
2 | ismnd.p | . . 3 | |
3 | 1, 2 | ismnddef 17296 | . 2 SGrp |
4 | rexn0 4074 | . . . 4 | |
5 | fvprc 6185 | . . . . . 6 | |
6 | 1, 5 | syl5eq 2668 | . . . . 5 |
7 | 6 | necon1ai 2821 | . . . 4 |
8 | 1, 2 | issgrpv 17286 | . . . 4 SGrp |
9 | 4, 7, 8 | 3syl 18 | . . 3 SGrp |
10 | 9 | pm5.32ri 670 | . 2 SGrp |
11 | 3, 10 | bitri 264 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wral 2912 wrex 2913 cvv 3200 c0 3915 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 SGrpcsgrp 17283 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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
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-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: mndid 17303 ismndd 17313 mndpropd 17316 mhmmnd 17537 signswmnd 30634 |
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