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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 33254. (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 19830 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
7 | 1, 2, 3, 6 | syl3anc 1370 | . . 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 33019 | . . . 4 ⊢ (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑍 + 𝑉) + (𝑈 + 𝑊))) |
13 | 1 | cmnmndd 19836 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
14 | 4, 5 | mndcl 18767 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑍 + 𝑉) ∈ 𝐵) |
15 | 13, 8, 10, 14 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑍 + 𝑉) ∈ 𝐵) |
16 | 4, 5 | mndcl 18767 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈 + 𝑊) ∈ 𝐵) |
17 | 13, 9, 11, 16 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑈 + 𝑊) ∈ 𝐵) |
18 | 4, 5 | cmncom 19830 | . . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ (𝑍 + 𝑉) ∈ 𝐵 ∧ (𝑈 + 𝑊) ∈ 𝐵) → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
19 | 1, 15, 17, 18 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
20 | 12, 19 | eqtrd 2774 | . . 3 ⊢ (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉))) |
21 | 7, 20 | oveq12d 7448 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉)))) |
22 | 4, 5, 1, 3, 2, 17, 15 | cmn4d 33019 | . 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉)))) |
23 | 21, 22 | eqtrd 2774 | 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 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Mndcmnd 18759 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: rlocaddval 33254 rlocmulval 33255 |
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