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Theorem abladdsub4 19599
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 19574 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1134 . . 3 ((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 18763 . . . 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 18763 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1372 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
135, 6grpcl 18763 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
142, 9, 4, 13syl3anc 1372 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
15 ablsubadd.m . . . 4 = (-g𝐺)
165, 15grpsubrcan 18835 . . 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 1373 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
18 simp1 1137 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
195, 6, 15ablsub4 19598 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
2018, 3, 4, 9, 4, 19syl122anc 1380 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
21 eqid 2737 . . . . . . 7 (0g𝐺) = (0g𝐺)
225, 21, 15grpsubid 18838 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 𝑌) = (0g𝐺))
232, 4, 22syl2anc 585 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑌) = (0g𝐺))
2423oveq2d 7378 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑌)) = ((𝑋 𝑍) + (0g𝐺)))
255, 15grpsubcl 18834 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
262, 3, 9, 25syl3anc 1372 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) ∈ 𝐵)
275, 6, 21grprid 18788 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
282, 26, 27syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
2920, 24, 283eqtrd 2781 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = (𝑋 𝑍))
305, 6, 15ablsub4 19598 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑊𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
3118, 9, 10, 9, 4, 30syl122anc 1380 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
325, 21, 15grpsubid 18838 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 𝑍) = (0g𝐺))
332, 9, 32syl2anc 585 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 𝑍) = (0g𝐺))
3433oveq1d 7377 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 𝑍) + (𝑊 𝑌)) = ((0g𝐺) + (𝑊 𝑌)))
355, 15grpsubcl 18834 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑊𝐵𝑌𝐵) → (𝑊 𝑌) ∈ 𝐵)
362, 10, 4, 35syl3anc 1372 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑊 𝑌) ∈ 𝐵)
375, 6, 21grplid 18787 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑊 𝑌) ∈ 𝐵) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
382, 36, 37syl2anc 585 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
3931, 34, 383eqtrd 2781 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = (𝑊 𝑌))
4029, 39eqeq12d 2753 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
4117, 40bitr3d 281 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  0gc0g 17328  Grpcgrp 18755  -gcsg 18757  Abelcabl 19570
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-cmn 19571  df-abl 19572
This theorem is referenced by:  lmodvaddsub4  20390
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