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Theorem ablpnpcan 18939
Description: Cancellation law for mixed addition and subtraction. (pnpcan 10924 analog.) (Contributed by NM, 29-May-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)
ablpnpcan.g (𝜑𝐺 ∈ Abel)
ablpnpcan.x (𝜑𝑋𝐵)
ablpnpcan.y (𝜑𝑌𝐵)
ablpnpcan.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablpnpcan (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))

Proof of Theorem ablpnpcan
StepHypRef Expression
1 ablsubsub.g . . 3 (𝜑𝐺 ∈ Abel)
2 ablsubsub.x . . 3 (𝜑𝑋𝐵)
3 ablsubsub.y . . 3 (𝜑𝑌𝐵)
4 ablsubsub.z . . 3 (𝜑𝑍𝐵)
5 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . 4 + = (+g𝐺)
7 ablsubadd.m . . . 4 = (-g𝐺)
85, 6, 7ablsub4 18932 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋𝐵𝑍𝐵)) → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
91, 2, 3, 2, 4, 8syl122anc 1375 . 2 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
10 ablgrp 18910 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
111, 10syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
12 eqid 2821 . . . . 5 (0g𝐺) = (0g𝐺)
135, 12, 7grpsubid 18182 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1411, 2, 13syl2anc 586 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1514oveq1d 7170 . 2 (𝜑 → ((𝑋 𝑋) + (𝑌 𝑍)) = ((0g𝐺) + (𝑌 𝑍)))
165, 7grpsubcl 18178 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1711, 3, 4, 16syl3anc 1367 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
185, 6, 12grplid 18132 . . 3 ((𝐺 ∈ Grp ∧ (𝑌 𝑍) ∈ 𝐵) → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
1911, 17, 18syl2anc 586 . 2 (𝜑 → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
209, 15, 193eqtrd 2860 1 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  cfv 6354  (class class class)co 7155  Basecbs 16482  +gcplusg 16564  0gc0g 16712  Grpcgrp 18102  -gcsg 18104  Abelcabl 18906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-0g 16714  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105  df-minusg 18106  df-sbg 18107  df-cmn 18907  df-abl 18908
This theorem is referenced by:  hdmaprnlem7N  38990
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