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Theorem abladdsub4 18533
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 18512 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
213ad2ant1 1164 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Grp)
3 simp2l 1257 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑋𝐵)
4 simp2r 1258 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑌𝐵)
5 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . . 5 + = (+g𝐺)
75, 6grpcl 17745 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1491 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
9 simp3l 1259 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑍𝐵)
10 simp3r 1260 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝑊𝐵)
115, 6grpcl 17745 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
122, 9, 10, 11syl3anc 1491 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑊) ∈ 𝐵)
135, 6grpcl 17745 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
142, 9, 4, 13syl3anc 1491 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
15 ablsubadd.m . . . 4 = (-g𝐺)
165, 15grpsubrcan 17811 . . 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 1492 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 + 𝑌) = (𝑍 + 𝑊)))
18 simp1 1167 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → 𝐺 ∈ Abel)
195, 6, 15ablsub4 18532 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
2018, 3, 4, 9, 4, 19syl122anc 1499 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑋 𝑍) + (𝑌 𝑌)))
21 eqid 2800 . . . . . . 7 (0g𝐺) = (0g𝐺)
225, 21, 15grpsubid 17814 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 𝑌) = (0g𝐺))
232, 4, 22syl2anc 580 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑌 𝑌) = (0g𝐺))
2423oveq2d 6895 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (𝑌 𝑌)) = ((𝑋 𝑍) + (0g𝐺)))
255, 15grpsubcl 17810 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
262, 3, 9, 25syl3anc 1491 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑋 𝑍) ∈ 𝐵)
275, 6, 21grprid 17768 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
282, 26, 27syl2anc 580 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑍) + (0g𝐺)) = (𝑋 𝑍))
2920, 24, 283eqtrd 2838 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑌)) = (𝑋 𝑍))
305, 6, 15ablsub4 18532 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑊𝐵) ∧ (𝑍𝐵𝑌𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
3118, 9, 10, 9, 4, 30syl122anc 1499 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = ((𝑍 𝑍) + (𝑊 𝑌)))
325, 21, 15grpsubid 17814 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 𝑍) = (0g𝐺))
332, 9, 32syl2anc 580 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑍 𝑍) = (0g𝐺))
3433oveq1d 6894 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 𝑍) + (𝑊 𝑌)) = ((0g𝐺) + (𝑊 𝑌)))
355, 15grpsubcl 17810 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑊𝐵𝑌𝐵) → (𝑊 𝑌) ∈ 𝐵)
362, 10, 4, 35syl3anc 1491 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (𝑊 𝑌) ∈ 𝐵)
375, 6, 21grplid 17767 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑊 𝑌) ∈ 𝐵) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
382, 36, 37syl2anc 580 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((0g𝐺) + (𝑊 𝑌)) = (𝑊 𝑌))
3931, 34, 383eqtrd 2838 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑍 + 𝑊) (𝑍 + 𝑌)) = (𝑊 𝑌))
4029, 39eqeq12d 2815 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → (((𝑋 + 𝑌) (𝑍 + 𝑌)) = ((𝑍 + 𝑊) (𝑍 + 𝑌)) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
4117, 40bitr3d 273 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  cfv 6102  (class class class)co 6879  Basecbs 16183  +gcplusg 16266  0gc0g 16414  Grpcgrp 17737  -gcsg 17739  Abelcabl 18508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-0g 16416  df-mgm 17556  df-sgrp 17598  df-mnd 17609  df-grp 17740  df-minusg 17741  df-sbg 17742  df-cmn 18509  df-abl 18510
This theorem is referenced by:  lmodvaddsub4  19232
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