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Theorem ablsubsub4 19785
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 19752 . . . . 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 18984 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1368 . . 3 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
10 ablsubsub.z . . 3 (𝜑𝑍𝐵)
11 ablsubadd.p . . . 4 + = (+g𝐺)
12 eqid 2725 . . . 4 (invg𝐺) = (invg𝐺)
136, 11, 12, 7grpsubval 18950 . . 3 (((𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
149, 10, 13syl2anc 582 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
156, 12grpinvcl 18952 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
163, 10, 15syl2anc 582 . . 3 (𝜑 → ((invg𝐺)‘𝑍) ∈ 𝐵)
176, 11, 7, 1, 4, 5, 16ablsubsub 19784 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
186, 11, 7, 12, 3, 5, 10grpsubinv 18976 . . 3 (𝜑 → (𝑌 ((invg𝐺)‘𝑍)) = (𝑌 + 𝑍))
1918oveq2d 7435 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = (𝑋 (𝑌 + 𝑍)))
2014, 17, 193eqtr2d 2771 1 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  Grpcgrp 18898  invgcminusg 18899  -gcsg 18900  Abelcabl 19748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-sbg 18903  df-cmn 19749  df-abl 19750
This theorem is referenced by:  ablsub32  19788  ablnnncan  19789  rngqiprngfulem4  21221  ip2subdi  21593  cpmadugsumlemF  22822  baerlem5alem2  41311
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