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Theorem abladdsub 18855
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
abladdsub ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))

Proof of Theorem abladdsub
StepHypRef Expression
1 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
2 ablsubadd.p . . . . 5 + = (+g𝐺)
31, 2ablcom 18844 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
433adant3r3 1178 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
54oveq1d 7163 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑌 + 𝑋) 𝑍))
6 ablgrp 18831 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
76adantr 481 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
8 simpr2 1189 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
9 simpr1 1188 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
10 simpr3 1190 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
11 ablsubadd.m . . . 4 = (-g𝐺)
121, 2, 11grpaddsubass 18119 . . 3 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵𝑍𝐵)) → ((𝑌 + 𝑋) 𝑍) = (𝑌 + (𝑋 𝑍)))
137, 8, 9, 10, 12syl13anc 1366 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 + 𝑋) 𝑍) = (𝑌 + (𝑋 𝑍)))
14 simpl 483 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Abel)
151, 11grpsubcl 18109 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
167, 9, 10, 15syl3anc 1365 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
171, 2ablcom 18844 . . 3 ((𝐺 ∈ Abel ∧ 𝑌𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → (𝑌 + (𝑋 𝑍)) = ((𝑋 𝑍) + 𝑌))
1814, 8, 16, 17syl3anc 1365 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 + (𝑋 𝑍)) = ((𝑋 𝑍) + 𝑌))
195, 13, 183eqtrd 2865 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  cfv 6352  (class class class)co 7148  Basecbs 16473  +gcplusg 16555  Grpcgrp 18033  -gcsg 18035  Abelcabl 18827
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-minusg 18037  df-sbg 18038  df-cmn 18828  df-abl 18829
This theorem is referenced by:  ablpncan2  18856  ablsubsub  18858  ip2subdi  20707
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