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

Theorem ablsubaddsub 19832
Description: Double subtraction and addition in abelian groups. (cnambpcma 47306 analog.) (Contributed by AV, 3-Mar-2025.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablsubaddsub ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = (𝑍 𝑌))

Proof of Theorem ablsubaddsub
StepHypRef Expression
1 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
2 ablsubadd.p . . . 4 + = (+g𝐺)
3 ablsubadd.m . . . 4 = (-g𝐺)
41, 2, 3ablsubadd23 19831 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑍) = (𝑋 + (𝑍 𝑌)))
54oveq1d 7446 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = ((𝑋 + (𝑍 𝑌)) 𝑋))
6 simpl 482 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Abel)
7 simpr1 1195 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
8 ablgrp 19803 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
98adantr 480 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr3 1197 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
11 simpr2 1196 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
121, 3grpsubcl 19038 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
139, 10, 11, 12syl3anc 1373 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑌) ∈ 𝐵)
141, 2ablcom 19817 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
156, 7, 13, 14syl3anc 1373 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
1615oveq1d 7446 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + (𝑍 𝑌)) 𝑋) = (((𝑍 𝑌) + 𝑋) 𝑋))
171, 2, 3grpaddsubass 19048 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑍 𝑌) ∈ 𝐵𝑋𝐵𝑋𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
189, 13, 7, 7, 17syl13anc 1374 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
19 eqid 2737 . . . . . 6 (0g𝐺) = (0g𝐺)
201, 19, 3grpsubid 19042 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
219, 7, 20syl2anc 584 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋) = (0g𝐺))
2221oveq2d 7447 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (𝑋 𝑋)) = ((𝑍 𝑌) + (0g𝐺)))
231, 2, 19, 9, 13grpridd 18988 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (0g𝐺)) = (𝑍 𝑌))
2418, 22, 233eqtrd 2781 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = (𝑍 𝑌))
255, 16, 243eqtrd 2781 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  -gcsg 18953  Abelcabl 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19800  df-abl 19801
This theorem is referenced by:  rngqiprngimfo  21311
  Copyright terms: Public domain W3C validator