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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 17520 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ SGrp) | |
2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | sgrpass 17511 | . 2 ⊢ ((𝐺 ∈ SGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
5 | 1, 4 | sylan 489 | 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 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 +gcplusg 16163 SGrpcsgrp 17504 Mndcmnd 17515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-nul 4941 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-iota 6012 df-fv 6057 df-ov 6817 df-sgrp 17505 df-mnd 17516 |
This theorem is referenced by: mnd32g 17526 mnd12g 17527 mnd4g 17528 issubmnd 17539 prdsmndd 17544 imasmnd 17549 mrcmndind 17587 gsumccat 17599 grpass 17652 mhmmnd 17758 mulgnndirOLD 17791 cntzsubm 17988 oppgmnd 18004 frgp0 18393 mulgnn0di 18451 gsumval3eu 18525 gsumval3 18528 srgass 18733 ringass 18784 mndvass 20420 chfacfscmulgsum 20887 chfacfpmmulgsum 20891 slmdass 30096 lidlmsgrp 42454 invginvrid 42676 |
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