| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mndassd | Structured version Visualization version GIF version | ||
| Description: A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| Ref | Expression |
|---|---|
| mndassd.1 | ⊢ 𝐵 = (Base‘𝐺) |
| mndassd.2 | ⊢ + = (+g‘𝐺) |
| mndassd.3 | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| mndassd.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| mndassd.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| mndassd.6 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| mndassd | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndassd.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
| 2 | mndassd.4 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | mndassd.5 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 4 | mndassd.6 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 5 | mndassd.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 6 | mndassd.2 | . . 3 ⊢ + = (+g‘𝐺) | |
| 7 | 5, 6 | mndass 18752 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 8 | 1, 2, 3, 4, 7 | syl13anc 1374 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 +gcplusg 17293 Mndcmnd 18743 |
| 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 2707 ax-nul 5304 |
| 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 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 df-ov 7432 df-sgrp 18728 df-mnd 18744 |
| This theorem is referenced by: mndlrinv 33014 mndlactf1 33016 mndlactfo 33017 mndractf1 33018 mndractfo 33019 mndlactf1o 33020 mndractf1o 33021 gsumwun 33053 |
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