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Mirrors > Home > MPE Home > Th. List > mndass | Structured version Visualization version Unicode version |
Description: A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.) |
Ref | Expression |
---|---|
mndcl.b | |
mndcl.p |
Ref | Expression |
---|---|
mndass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndsgrp 17299 | . 2 SGrp | |
2 | mndcl.b | . . 3 | |
3 | mndcl.p | . . 3 | |
4 | 2, 3 | sgrpass 17290 | . 2 SGrp |
5 | 1, 4 | sylan 488 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 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-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-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: mnd32g 17305 mnd12g 17306 mnd4g 17307 issubmnd 17318 prdsmndd 17323 imasmnd 17328 mrcmndind 17366 gsumccat 17378 grpass 17431 mhmmnd 17537 mulgnndirOLD 17570 cntzsubm 17768 oppgmnd 17784 frgp0 18173 mulgnn0di 18231 gsumval3eu 18305 gsumval3 18308 srgass 18513 ringass 18564 mndvass 20198 chfacfscmulgsum 20665 chfacfpmmulgsum 20669 slmdass 29766 lidlmsgrp 41926 invginvrid 42148 |
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