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Theorem cmn4 19834
Description: Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
Assertion
Ref Expression
cmn4 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Proof of Theorem cmn4
StepHypRef Expression
1 ablcom.b . 2 𝐵 = (Base‘𝐺)
2 ablcom.p . 2 + = (+g𝐺)
3 simp1 1135 . . 3 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
4 cmnmnd 19830 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53, 4syl 17 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Mnd)
6 simp2l 1198 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
7 simp2r 1199 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
8 simp3l 1200 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
9 simp3r 1201 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
101, 2cmncom 19831 . . 3 ((𝐺 ∈ CMnd ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
113, 7, 8, 10syl3anc 1370 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
121, 2, 5, 6, 7, 8, 9, 11mnd4g 18774 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Mndcmnd 18760  CMndccmn 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-cmn 19815
This theorem is referenced by:  ablsub4  19843  ghmplusg  19879  lmod4  20927  ip2di  21677  evlslem1  22124  clmsub4  25153  cmn4d  33020  cringm4  33454  lfladdcl  39053
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