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Theorem ablsub4 19001
 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 18978 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1130 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1196 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1197 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 18177 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1368 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1198 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1199 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 18177 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1368 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
13 eqid 2758 . . . 4 (invg𝐺) = (invg𝐺)
14 ablsubadd.m . . . 4 = (-g𝐺)
155, 6, 13, 14grpsubval 18216 . . 3 (((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
168, 12, 15syl2anc 587 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
17 ablcmn 18980 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
18173ad2ant1 1130 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
19 simp2 1134 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋𝐵𝑌𝐵))
205, 13grpinvcl 18218 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
212, 9, 20syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
225, 13grpinvcl 18218 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑊𝐵) → ((invg𝐺)‘𝑊) ∈ 𝐵)
232, 10, 22syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑊) ∈ 𝐵)
245, 6cmn4 18993 . . . 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 1371 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
26 simp1 1133 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
275, 6, 13ablinvadd 18998 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑍𝐵𝑊𝐵) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2826, 9, 10, 27syl3anc 1368 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2928oveq2d 7166 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))))
305, 6, 13, 14grpsubval 18216 . . . . 5 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
313, 9, 30syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
325, 6, 13, 14grpsubval 18216 . . . . 5 ((𝑌𝐵𝑊𝐵) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
334, 10, 32syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
3431, 33oveq12d 7168 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑊)) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
3525, 29, 343eqtr4d 2803 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 𝑍) + (𝑌 𝑊)))
3616, 35eqtrd 2793 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  Grpcgrp 18169  invgcminusg 18170  -gcsg 18171  CMndccmn 18973  Abelcabl 18974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-0g 16773  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-minusg 18173  df-sbg 18174  df-cmn 18975  df-abl 18976 This theorem is referenced by:  abladdsub4  19002  ablpnpcan  19008  mdetuni0  21321  minveclem2  24126  baerlem3lem1  39283
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