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Theorem cmn246135 33020
Description: Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33254. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
cmn135246.1 𝐵 = (Base‘𝐺)
cmn135246.2 + = (+g𝐺)
cmn135246.3 (𝜑𝐺 ∈ CMnd)
cmn135246.5 (𝜑𝑋𝐵)
cmn135246.4 (𝜑𝑌𝐵)
cmn135246.6 (𝜑𝑍𝐵)
cmn135246.7 (𝜑𝑈𝐵)
cmn135246.8 (𝜑𝑉𝐵)
cmn135246.9 (𝜑𝑊𝐵)
Assertion
Ref Expression
cmn246135 (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉))))

Proof of Theorem cmn246135
StepHypRef Expression
1 cmn135246.3 . . . 4 (𝜑𝐺 ∈ CMnd)
2 cmn135246.5 . . . 4 (𝜑𝑋𝐵)
3 cmn135246.4 . . . 4 (𝜑𝑌𝐵)
4 cmn135246.1 . . . . 5 𝐵 = (Base‘𝐺)
5 cmn135246.2 . . . . 5 + = (+g𝐺)
64, 5cmncom 19830 . . . 4 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
71, 2, 3, 6syl3anc 1370 . . 3 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
8 cmn135246.6 . . . . 5 (𝜑𝑍𝐵)
9 cmn135246.7 . . . . 5 (𝜑𝑈𝐵)
10 cmn135246.8 . . . . 5 (𝜑𝑉𝐵)
11 cmn135246.9 . . . . 5 (𝜑𝑊𝐵)
124, 5, 1, 8, 9, 10, 11cmn4d 33019 . . . 4 (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑍 + 𝑉) + (𝑈 + 𝑊)))
131cmnmndd 19836 . . . . . 6 (𝜑𝐺 ∈ Mnd)
144, 5mndcl 18767 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑉𝐵) → (𝑍 + 𝑉) ∈ 𝐵)
1513, 8, 10, 14syl3anc 1370 . . . . 5 (𝜑 → (𝑍 + 𝑉) ∈ 𝐵)
164, 5mndcl 18767 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑈𝐵𝑊𝐵) → (𝑈 + 𝑊) ∈ 𝐵)
1713, 9, 11, 16syl3anc 1370 . . . . 5 (𝜑 → (𝑈 + 𝑊) ∈ 𝐵)
184, 5cmncom 19830 . . . . 5 ((𝐺 ∈ CMnd ∧ (𝑍 + 𝑉) ∈ 𝐵 ∧ (𝑈 + 𝑊) ∈ 𝐵) → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉)))
191, 15, 17, 18syl3anc 1370 . . . 4 (𝜑 → ((𝑍 + 𝑉) + (𝑈 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉)))
2012, 19eqtrd 2774 . . 3 (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑈 + 𝑊) + (𝑍 + 𝑉)))
217, 20oveq12d 7448 . 2 (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉))))
224, 5, 1, 3, 2, 17, 15cmn4d 33019 . 2 (𝜑 → ((𝑌 + 𝑋) + ((𝑈 + 𝑊) + (𝑍 + 𝑉))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉))))
2321, 22eqtrd 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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