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Theorem ablsub4 18041
 Description: Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
ablsub4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))

Proof of Theorem ablsub4
StepHypRef Expression
1 ablgrp 18021 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1075 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1080 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1081 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 17253 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1318 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1082 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1083 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 17253 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1318 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
13 eqid 2610 . . . 4 (invg𝐺) = (invg𝐺)
14 ablsubadd.m . . . 4 = (-g𝐺)
155, 6, 13, 14grpsubval 17288 . . 3 (((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
168, 12, 15syl2anc 691 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
17 ablcmn 18022 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
18173ad2ant1 1075 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
19 simp2 1055 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋𝐵𝑌𝐵))
205, 13grpinvcl 17290 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
212, 9, 20syl2anc 691 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
225, 13grpinvcl 17290 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑊𝐵) → ((invg𝐺)‘𝑊) ∈ 𝐵)
232, 10, 22syl2anc 691 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑊) ∈ 𝐵)
245, 6cmn4 18035 . . . 4 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (((invg𝐺)‘𝑍) ∈ 𝐵 ∧ ((invg𝐺)‘𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
2518, 19, 21, 23, 24syl112anc 1322 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
26 simp1 1054 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
275, 6, 13ablinvadd 18038 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑍𝐵𝑊𝐵) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2826, 9, 10, 27syl3anc 1318 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2928oveq2d 6565 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))))
305, 6, 13, 14grpsubval 17288 . . . . 5 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
313, 9, 30syl2anc 691 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
325, 6, 13, 14grpsubval 17288 . . . . 5 ((𝑌𝐵𝑊𝐵) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
334, 10, 32syl2anc 691 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
3431, 33oveq12d 6567 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑊)) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
3525, 29, 343eqtr4d 2654 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 𝑍) + (𝑌 𝑊)))
3616, 35eqtrd 2644 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247  CMndccmn 18016  Abelcabl 18017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-cmn 18018  df-abl 18019 This theorem is referenced by:  abladdsub4  18042  ablpnpcan  18048  mdetuni0  20246  minveclem2  23005  baerlem3lem1  36014
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