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Theorem cmn4 19768
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 1133 . . 3 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
4 cmnmnd 19764 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53, 4syl 17 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Mnd)
6 simp2l 1196 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
7 simp2r 1197 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
8 simp3l 1198 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
9 simp3r 1199 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
101, 2cmncom 19765 . . 3 ((𝐺 ∈ CMnd ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
113, 7, 8, 10syl3anc 1368 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
121, 2, 5, 6, 7, 8, 9, 11mnd4g 18711 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  Mndcmnd 18697  CMndccmn 19747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-cmn 19749
This theorem is referenced by:  ablsub4  19777  ghmplusg  19813  lmod4  20807  ip2di  21590  evlslem1  22050  clmsub4  25077  cmn4d  32841  cringm4  33258  lfladdcl  38670
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