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Theorem ablsub4 19678
Description: Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablsub4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))

Proof of Theorem ablsub4
StepHypRef Expression
1 ablgrp 19653 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1134 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1200 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1201 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 18827 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1372 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1202 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1203 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 18827 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1372 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
13 eqid 2733 . . . 4 (invg𝐺) = (invg𝐺)
14 ablsubadd.m . . . 4 = (-g𝐺)
155, 6, 13, 14grpsubval 18870 . . 3 (((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
168, 12, 15syl2anc 585 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))))
17 ablcmn 19655 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
18173ad2ant1 1134 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ CMnd)
19 simp2 1138 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋𝐵𝑌𝐵))
205, 13grpinvcl 18872 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
212, 9, 20syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
225, 13grpinvcl 18872 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑊𝐵) → ((invg𝐺)‘𝑊) ∈ 𝐵)
232, 10, 22syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘𝑊) ∈ 𝐵)
245, 6cmn4 19669 . . . 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 1375 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
26 simp1 1137 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
275, 6, 13ablinvadd 19675 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑍𝐵𝑊𝐵) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2826, 9, 10, 27syl3anc 1372 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((invg𝐺)‘(𝑍 + 𝑊)) = (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊)))
2928oveq2d 7425 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 + 𝑌) + (((invg𝐺)‘𝑍) + ((invg𝐺)‘𝑊))))
305, 6, 13, 14grpsubval 18870 . . . . 5 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
313, 9, 30syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) = (𝑋 + ((invg𝐺)‘𝑍)))
325, 6, 13, 14grpsubval 18870 . . . . 5 ((𝑌𝐵𝑊𝐵) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
334, 10, 32syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑊) = (𝑌 + ((invg𝐺)‘𝑊)))
3431, 33oveq12d 7427 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑊)) = ((𝑋 + ((invg𝐺)‘𝑍)) + (𝑌 + ((invg𝐺)‘𝑊))))
3525, 29, 343eqtr4d 2783 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘(𝑍 + 𝑊))) = ((𝑋 𝑍) + (𝑌 𝑊)))
3616, 35eqtrd 2773 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Grpcgrp 18819  invgcminusg 18820  -gcsg 18821  CMndccmn 19648  Abelcabl 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-cmn 19650  df-abl 19651
This theorem is referenced by:  abladdsub4  19679  ablpnpcan  19687  mdetuni0  22123  minveclem2  24943  baerlem3lem1  40626
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