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Theorem grpsubsub 18126
Description: Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpsubsub ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑋 + (𝑍 𝑌)))

Proof of Theorem grpsubsub
StepHypRef Expression
1 simpr1 1186 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
2 grpsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpsubadd.m . . . . 5 = (-g𝐺)
42, 3grpsubcl 18117 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
543adant3r1 1174 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
6 grpsubadd.p . . . 4 + = (+g𝐺)
7 eqid 2818 . . . 4 (invg𝐺) = (invg𝐺)
82, 6, 7, 3grpsubval 18087 . . 3 ((𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = (𝑋 + ((invg𝐺)‘(𝑌 𝑍))))
91, 5, 8syl2anc 584 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑋 + ((invg𝐺)‘(𝑌 𝑍))))
102, 3, 7grpinvsub 18119 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → ((invg𝐺)‘(𝑌 𝑍)) = (𝑍 𝑌))
11103adant3r1 1174 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘(𝑌 𝑍)) = (𝑍 𝑌))
1211oveq2d 7161 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘(𝑌 𝑍))) = (𝑋 + (𝑍 𝑌)))
139, 12eqtrd 2853 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑋 + (𝑍 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Grpcgrp 18041  invgcminusg 18042  -gcsg 18043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046
This theorem is referenced by:  ablsubsub  18867
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