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Theorem grpaddsubass 17445
 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 473 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 simpr1 1065 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
3 simpr2 1066 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
4 grpsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
5 eqid 2621 . . . . 5 (invg𝐺) = (invg𝐺)
64, 5grpinvcl 17407 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
763ad2antr3 1226 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
8 grpsubadd.p . . . 4 + = (+g𝐺)
94, 8grpass 17371 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
101, 2, 3, 7, 9syl13anc 1325 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
114, 8grpcl 17370 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
12113adant3r3 1273 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) ∈ 𝐵)
13 simpr3 1067 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
14 grpsubadd.m . . . 4 = (-g𝐺)
154, 8, 5, 14grpsubval 17405 . . 3 (((𝑋 + 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)))
1612, 13, 15syl2anc 692 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 + 𝑌) + ((invg𝐺)‘𝑍)))
174, 8, 5, 14grpsubval 17405 . . . 4 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌 + ((invg𝐺)‘𝑍)))
183, 13, 17syl2anc 692 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) = (𝑌 + ((invg𝐺)‘𝑍)))
1918oveq2d 6631 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 𝑍)) = (𝑋 + (𝑌 + ((invg𝐺)‘𝑍))))
2010, 16, 193eqtr4d 2665 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = (𝑋 + (𝑌 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Grpcgrp 17362  invgcminusg 17363  -gcsg 17364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367 This theorem is referenced by:  grppncan  17446  grpnpncan  17450  nsgconj  17567  conjghm  17631  conjnmz  17634  conjnmzb  17635  sylow3lem1  17982  sylow3lem2  17983  abladdsub  18160  ablsubsub  18163  cpmadugsumlemF  20621  archiabllem2a  29575
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