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Theorem abladdsub4 19232
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 19208 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1135 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1201 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1202 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 18406 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1373 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1203 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1204 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 18406 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1373 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
135, 6grpcl 18406 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
142, 9, 4, 13syl3anc 1373 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
15 ablsubadd.m . . . 4 = (-g𝐺)
165, 15grpsubrcan 18477 . . 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 1374 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
18 simp1 1138 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
195, 6, 15ablsub4 19231 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
2018, 3, 4, 9, 4, 19syl122anc 1381 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
21 eqid 2739 . . . . . . 7 (0g𝐺) = (0g𝐺)
225, 21, 15grpsubid 18480 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 𝑌) = (0g𝐺))
232, 4, 22syl2anc 587 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑌) = (0g𝐺))
2423oveq2d 7251 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑌)) = ((𝑋 𝑍) + (0g𝐺)))
255, 15grpsubcl 18476 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
262, 3, 9, 25syl3anc 1373 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) ∈ 𝐵)
275, 6, 21grprid 18431 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
282, 26, 27syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
2920, 24, 283eqtrd 2783 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = (𝑋 𝑍))
305, 6, 15ablsub4 19231 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑊𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
3118, 9, 10, 9, 4, 30syl122anc 1381 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
325, 21, 15grpsubid 18480 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 𝑍) = (0g𝐺))
332, 9, 32syl2anc 587 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 𝑍) = (0g𝐺))
3433oveq1d 7250 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 𝑍) + (𝑊 𝑌)) = ((0g𝐺) + (𝑊 𝑌)))
355, 15grpsubcl 18476 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑊𝐵𝑌𝐵) → (𝑊 𝑌) ∈ 𝐵)
362, 10, 4, 35syl3anc 1373 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑊 𝑌) ∈ 𝐵)
375, 6, 21grplid 18430 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑊 𝑌) ∈ 𝐵) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
382, 36, 37syl2anc 587 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
3931, 34, 383eqtrd 2783 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = (𝑊 𝑌))
4029, 39eqeq12d 2755 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
4117, 40bitr3d 284 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2112  cfv 6401  (class class class)co 7235  Basecbs 16793  +gcplusg 16835  0gc0g 16977  Grpcgrp 18398  -gcsg 18400  Abelcabl 19204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-id 5472  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-fv 6409  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-1st 7783  df-2nd 7784  df-0g 16979  df-mgm 18147  df-sgrp 18196  df-mnd 18207  df-grp 18401  df-minusg 18402  df-sbg 18403  df-cmn 19205  df-abl 19206
This theorem is referenced by:  lmodvaddsub4  19984
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