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Theorem cmn4 18136
Description: Commutative/associative law for Abelian groups. (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 1059 . . 3 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
4 cmnmnd 18132 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53, 4syl 17 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Mnd)
6 simp2l 1085 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
7 simp2r 1086 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
8 simp3l 1087 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
9 simp3r 1088 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
101, 2cmncom 18133 . . 3 ((𝐺 ∈ CMnd ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
113, 7, 8, 10syl3anc 1323 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
121, 2, 5, 6, 7, 8, 9, 11mnd4g 17231 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5849  (class class class)co 6607  Basecbs 15784  +gcplusg 15865  Mndcmnd 17218  CMndccmn 18117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-iota 5812  df-fv 5857  df-ov 6610  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-cmn 18119
This theorem is referenced by:  ablsub4  18142  ghmplusg  18173  lmod4  18837  evlslem1  19437  ip2di  19908  clmsub4  22819  lfladdcl  33859
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