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Theorem cmn145236 33021
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
cmn145236 (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 𝑊))))

Proof of Theorem cmn145236
StepHypRef Expression
1 cmn135246.3 . . . . . 6 (𝜑𝐺 ∈ CMnd)
2 cmn135246.6 . . . . . 6 (𝜑𝑍𝐵)
3 cmn135246.7 . . . . . 6 (𝜑𝑈𝐵)
4 cmn135246.1 . . . . . . 7 𝐵 = (Base‘𝐺)
5 cmn135246.2 . . . . . . 7 + = (+g𝐺)
64, 5cmncom 19830 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝑍𝐵𝑈𝐵) → (𝑍 + 𝑈) = (𝑈 + 𝑍))
71, 2, 3, 6syl3anc 1370 . . . . 5 (𝜑 → (𝑍 + 𝑈) = (𝑈 + 𝑍))
87oveq1d 7445 . . . 4 (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑈 + 𝑍) + (𝑉 + 𝑊)))
9 cmn135246.8 . . . . 5 (𝜑𝑉𝐵)
10 cmn135246.9 . . . . 5 (𝜑𝑊𝐵)
114, 5, 1, 3, 2, 9, 10cmn4d 33019 . . . 4 (𝜑 → ((𝑈 + 𝑍) + (𝑉 + 𝑊)) = ((𝑈 + 𝑉) + (𝑍 + 𝑊)))
128, 11eqtrd 2774 . . 3 (𝜑 → ((𝑍 + 𝑈) + (𝑉 + 𝑊)) = ((𝑈 + 𝑉) + (𝑍 + 𝑊)))
1312oveq2d 7446 . 2 (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + 𝑌) + ((𝑈 + 𝑉) + (𝑍 + 𝑊))))
14 cmn135246.5 . . 3 (𝜑𝑋𝐵)
151cmnmndd 19836 . . . 4 (𝜑𝐺 ∈ Mnd)
164, 5mndcl 18767 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑈𝐵𝑉𝐵) → (𝑈 + 𝑉) ∈ 𝐵)
1715, 3, 9, 16syl3anc 1370 . . 3 (𝜑 → (𝑈 + 𝑉) ∈ 𝐵)
18 cmn135246.4 . . 3 (𝜑𝑌𝐵)
194, 5mndcl 18767 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
2015, 2, 10, 19syl3anc 1370 . . 3 (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
214, 5, 1, 14, 17, 18, 20cmn4d 33019 . 2 (𝜑 → ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 𝑊))) = ((𝑋 + 𝑌) + ((𝑈 + 𝑉) + (𝑍 + 𝑊))))
2213, 21eqtr4d 2777 1 (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 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  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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