MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abladdsub4 Structured version   Visualization version   GIF version

Theorem abladdsub4 19844
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 19818 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1132 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1198 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1199 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 18972 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1200 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1201 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 18972 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
135, 6grpcl 18972 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
142, 9, 4, 13syl3anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
15 ablsubadd.m . . . 4 = (-g𝐺)
165, 15grpsubrcan 19052 . . 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 1371 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
18 simp1 1135 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
195, 6, 15ablsub4 19843 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
2018, 3, 4, 9, 4, 19syl122anc 1378 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
21 eqid 2735 . . . . . . 7 (0g𝐺) = (0g𝐺)
225, 21, 15grpsubid 19055 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 𝑌) = (0g𝐺))
232, 4, 22syl2anc 584 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑌) = (0g𝐺))
2423oveq2d 7447 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑌)) = ((𝑋 𝑍) + (0g𝐺)))
255, 15grpsubcl 19051 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
262, 3, 9, 25syl3anc 1370 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) ∈ 𝐵)
275, 6, 21grprid 18999 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
282, 26, 27syl2anc 584 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
2920, 24, 283eqtrd 2779 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = (𝑋 𝑍))
305, 6, 15ablsub4 19843 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑊𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
3118, 9, 10, 9, 4, 30syl122anc 1378 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
325, 21, 15grpsubid 19055 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 𝑍) = (0g𝐺))
332, 9, 32syl2anc 584 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 𝑍) = (0g𝐺))
3433oveq1d 7446 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 𝑍) + (𝑊 𝑌)) = ((0g𝐺) + (𝑊 𝑌)))
355, 15grpsubcl 19051 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑊𝐵𝑌𝐵) → (𝑊 𝑌) ∈ 𝐵)
362, 10, 4, 35syl3anc 1370 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑊 𝑌) ∈ 𝐵)
375, 6, 21grplid 18998 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑊 𝑌) ∈ 𝐵) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
382, 36, 37syl2anc 584 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
3931, 34, 383eqtrd 2779 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = (𝑊 𝑌))
4029, 39eqeq12d 2751 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  Abelcabl 19814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-cmn 19815  df-abl 19816
This theorem is referenced by:  lmodvaddsub4  20929
  Copyright terms: Public domain W3C validator