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Theorem ablsubadd 19729
Description: Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablsubadd ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))

Proof of Theorem ablsubadd
StepHypRef Expression
1 ablgrp 19705 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
3 ablsubadd.p . . . 4 + = (+g𝐺)
4 ablsubadd.m . . . 4 = (-g𝐺)
52, 3, 4grpsubadd 18956 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
61, 5sylan 579 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
72, 3ablcom 19719 . . . 4 ((𝐺 ∈ Abel ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
873adant3r1 1179 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
98eqeq1d 2728 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 + 𝑍) = 𝑋 ↔ (𝑍 + 𝑌) = 𝑋))
106, 9bitr4d 282 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863  -gcsg 18865  Abelcabl 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-cmn 19702  df-abl 19703
This theorem is referenced by:  lmodvsubadd  20759
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