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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmn4d | Structured version Visualization version GIF version |
Description: Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025.) |
Ref | Expression |
---|---|
cmn4d.1 | ⊢ 𝐵 = (Base‘𝐺) |
cmn4d.2 | ⊢ + = (+g‘𝐺) |
cmn4d.3 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
cmn4d.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
cmn4d.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
cmn4d.6 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
cmn4d.7 | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
Ref | Expression |
---|---|
cmn4d | ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmn4d.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
2 | cmn4d.4 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | cmn4d.5 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | cmn4d.6 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | cmn4d.7 | . 2 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
6 | cmn4d.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
7 | cmn4d.2 | . . 3 ⊢ + = (+g‘𝐺) | |
8 | 6, 7 | cmn4 19833 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
9 | 1, 2, 3, 4, 5, 8 | syl122anc 1378 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-cmn 19814 |
This theorem is referenced by: cmn246135 33020 cmn145236 33021 rloccring 33256 |
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