![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 18757 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
8 | 1, 2, 3, 4, 7 | syl13anc 1370 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 +gcplusg 17287 Mndcmnd 18748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-iota 6510 df-fv 6566 df-ov 7428 df-sgrp 18733 df-mnd 18749 |
This theorem is referenced by: mndlrinv 32988 mndlactf1 32990 mndlactfo 32991 mndractf1 32992 mndractfo 32993 mndlactf1o 32994 mndractf1o 32995 |
Copyright terms: Public domain | W3C validator |