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Theorem ablsubsub4 18868
Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
ablsubsub.g (𝜑𝐺 ∈ Abel)
ablsubsub.x (𝜑𝑋𝐵)
ablsubsub.y (𝜑𝑌𝐵)
ablsubsub.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablsubsub4 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))

Proof of Theorem ablsubsub4
StepHypRef Expression
1 ablsubsub.g . . . . 5 (𝜑𝐺 ∈ Abel)
2 ablgrp 18840 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
31, 2syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
4 ablsubsub.x . . . 4 (𝜑𝑋𝐵)
5 ablsubsub.y . . . 4 (𝜑𝑌𝐵)
6 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
7 ablsubadd.m . . . . 5 = (-g𝐺)
86, 7grpsubcl 18117 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1363 . . 3 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
10 ablsubsub.z . . 3 (𝜑𝑍𝐵)
11 ablsubadd.p . . . 4 + = (+g𝐺)
12 eqid 2818 . . . 4 (invg𝐺) = (invg𝐺)
136, 11, 12, 7grpsubval 18087 . . 3 (((𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
149, 10, 13syl2anc 584 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
156, 12grpinvcl 18089 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
163, 10, 15syl2anc 584 . . 3 (𝜑 → ((invg𝐺)‘𝑍) ∈ 𝐵)
176, 11, 7, 1, 4, 5, 16ablsubsub 18867 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
186, 11, 7, 12, 3, 5, 10grpsubinv 18110 . . 3 (𝜑 → (𝑌 ((invg𝐺)‘𝑍)) = (𝑌 + 𝑍))
1918oveq2d 7161 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = (𝑋 (𝑌 + 𝑍)))
2014, 17, 193eqtr2d 2859 1 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Grpcgrp 18041  invgcminusg 18042  -gcsg 18043  Abelcabl 18836
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  df-cmn 18837  df-abl 18838
This theorem is referenced by:  ablsub32  18871  ablnnncan  18872  ip2subdi  20716  cpmadugsumlemF  21412  baerlem5alem2  38727
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