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Theorem abladdsub4 18135
Description: Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
abladdsub4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))

Proof of Theorem abladdsub4
StepHypRef Expression
1 ablgrp 18114 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1080 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1085 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1086 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 17346 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1323 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1087 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1088 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 17346 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1323 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
135, 6grpcl 17346 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
142, 9, 4, 13syl3anc 1323 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
15 ablsubadd.m . . . 4 = (-g𝐺)
165, 15grpsubrcan 17412 . . 3 ((𝐺 ∈ Grp ∧ ((𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵 ∧ (𝑍 + 𝑌) ∈ 𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
172, 8, 12, 14, 16syl13anc 1325 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
18 simp1 1059 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
195, 6, 15ablsub4 18134 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
2018, 3, 4, 9, 4, 19syl122anc 1332 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
21 eqid 2626 . . . . . . 7 (0g𝐺) = (0g𝐺)
225, 21, 15grpsubid 17415 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 𝑌) = (0g𝐺))
232, 4, 22syl2anc 692 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑌) = (0g𝐺))
2423oveq2d 6621 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑌)) = ((𝑋 𝑍) + (0g𝐺)))
255, 15grpsubcl 17411 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
262, 3, 9, 25syl3anc 1323 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) ∈ 𝐵)
275, 6, 21grprid 17369 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
282, 26, 27syl2anc 692 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
2920, 24, 283eqtrd 2664 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = (𝑋 𝑍))
305, 6, 15ablsub4 18134 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑊𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
3118, 9, 10, 9, 4, 30syl122anc 1332 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
325, 21, 15grpsubid 17415 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 𝑍) = (0g𝐺))
332, 9, 32syl2anc 692 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 𝑍) = (0g𝐺))
3433oveq1d 6620 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 𝑍) + (𝑊 𝑌)) = ((0g𝐺) + (𝑊 𝑌)))
355, 15grpsubcl 17411 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑊𝐵𝑌𝐵) → (𝑊 𝑌) ∈ 𝐵)
362, 10, 4, 35syl3anc 1323 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑊 𝑌) ∈ 𝐵)
375, 6, 21grplid 17368 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑊 𝑌) ∈ 𝐵) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
382, 36, 37syl2anc 692 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
3931, 34, 383eqtrd 2664 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = (𝑊 𝑌))
4029, 39eqeq12d 2641 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
4117, 40bitr3d 270 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Grpcgrp 17338  -gcsg 17340  Abelcabl 18110
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-cmn 18111  df-abl 18112
This theorem is referenced by:  lmodvaddsub4  18831
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