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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cmn246135 | Structured version Visualization version GIF version | ||
| Description: Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33272. (Contributed by Thierry Arnoux, 4-May-2025.) |
| Ref | Expression |
|---|---|
| cmn135246.1 | ⊢ 𝐵 = (Base‘𝐺) |
| cmn135246.2 | ⊢ + = (+g‘𝐺) |
| cmn135246.3 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| cmn135246.5 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| cmn135246.4 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| cmn135246.6 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
| cmn135246.7 | ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
| cmn135246.8 | ⊢ (𝜑 → 𝑉 ∈ 𝐵) |
| cmn135246.9 | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| cmn246135 | ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cmn135246.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 2 | cmn135246.5 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | cmn135246.4 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 4 | cmn135246.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
| 5 | cmn135246.2 | . . . . 5 ⊢ + = (+g‘𝐺) | |
| 6 | 4, 5 | cmncom 19816 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| 7 | 1, 2, 3, 6 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| 8 | cmn135246.6 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 9 | cmn135246.7 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐵) | |
| 10 | cmn135246.8 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵) | |
| 11 | cmn135246.9 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
| 12 | 4, 5, 1, 8, 9, 10, 11 | cmn4d 33037 | . . . 4 ⊢ (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑍 + 𝑉) + (𝑈 + 𝑊))) |
| 13 | 1 | cmnmndd 19822 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 14 | 4, 5 | mndcl 18755 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑍 + 𝑉) ∈ 𝐵) |
| 15 | 13, 8, 10, 14 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑍 + 𝑉) ∈ 𝐵) |
| 16 | 4, 5 | mndcl 18755 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈 + 𝑊) ∈ 𝐵) |
| 17 | 13, 9, 11, 16 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑈 + 𝑊) ∈ 𝐵) |
| 18 | 4, 5 | cmncom 19816 | . . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ (𝑍 + 𝑉) ∈ 𝐵 ∧ (𝑈 + 𝑊) ∈ 𝐵) → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
| 19 | 1, 15, 17, 18 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
| 20 | 12, 19 | eqtrd 2777 | . . 3 ⊢ (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
| 21 | 7, 20 | oveq12d 7449 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉)))) |
| 22 | 4, 5, 1, 3, 2, 17, 15 | cmn4d 33037 | . 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉)))) |
| 23 | 21, 22 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉)))) |
| 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 Mndcmnd 18747 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: rlocaddval 33272 rlocmulval 33273 |
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