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

Theorem grpaddsubass 19048
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpaddsubass ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = (𝑋 + (𝑌 𝑍)))

Proof of Theorem grpaddsubass
StepHypRef Expression
1 simpl 482 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 simpr1 1195 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
3 simpr2 1196 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
4 grpsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
5 eqid 2737 . . . . 5 (invg𝐺) = (invg𝐺)
64, 5grpinvcl 19005 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
763ad2antr3 1191 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
8 grpsubadd.p . . . 4 + = (+g𝐺)
94, 8grpass 18960 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
101, 2, 3, 7, 9syl13anc 1374 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
114, 8grpcl 18959 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
12113adant3r3 1185 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
13 simpr3 1197 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
14 grpsubadd.m . . . 4 = (-g𝐺)
154, 8, 5, 14grpsubval 19003 . . 3 (((𝑋 + 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)))
1612, 13, 15syl2anc 584 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)))
174, 8, 5, 14grpsubval 19003 . . . 4 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌 + ((invg𝐺)‘𝑍)))
183, 13, 17syl2anc 584 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) = (𝑌 + ((invg𝐺)‘𝑍)))
1918oveq2d 7447 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
2010, 16, 193eqtr4d 2787 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = (𝑋 + (𝑌 𝑍)))
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  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953
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
This theorem is referenced by:  grppncan  19049  grpnpncan  19053  nsgconj  19177  conjghm  19267  conjnmz  19270  conjnmzb  19271  sylow3lem1  19645  sylow3lem2  19646  abladdsub  19830  ablsubadd23  19831  ablsubaddsub  19832  ablsubsub  19835  cpmadugsumlemF  22882  archiabllem2a  33201
  Copyright terms: Public domain W3C validator