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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 19819 | . 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 1381 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 CMndccmn 19798 |
| 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-12 2177 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-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cmn 19800 |
| This theorem is referenced by: cmn246135 33038 cmn145236 33039 rloccring 33274 |
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