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| Mirrors > Home > MPE Home > Th. List > mndass | Structured version Visualization version GIF version | ||
| Description: A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.) |
| Ref | Expression |
|---|---|
| mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
| mndcl.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| mndass | ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndsgrp 18753 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) | |
| 2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 4 | 2, 3 | sgrpass 18738 | . 2 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 5 | 1, 4 | sylan 580 | 1 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Smgrpcsgrp 18731 Mndcmnd 18747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-sgrp 18732 df-mnd 18748 |
| This theorem is referenced by: mnd32g 18759 mnd12g 18760 mnd4g 18761 issubmnd 18774 mndinvmod 18777 prdsmndd 18783 imasmnd 18788 mndvass 18811 mndind 18841 grpass 18960 mhmmnd 19082 cntzsubm 19356 oppgmnd 19373 frgp0 19778 mulgnn0di 19843 gsumval3eu 19922 gsumval3 19925 srgass 20191 srgcom4 20211 ringass 20250 chfacfscmulgsum 22866 chfacfpmmulgsum 22870 mndassd 33028 slmdass 33219 lsmssass 33430 mndmolinv 42096 primrootsunit1 42098 invginvrid 48283 mndtccatid 49184 |
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