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Theorem ablpnpcan 19778
Description: Cancellation law for mixed addition and subtraction. (pnpcan 11529 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 19769 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋𝐵𝑍𝐵)) → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
91, 2, 3, 2, 4, 8syl122anc 1376 . 2 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
10 ablgrp 19744 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
111, 10syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
12 eqid 2725 . . . . 5 (0g𝐺) = (0g𝐺)
135, 12, 7grpsubid 18984 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1411, 2, 13syl2anc 582 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1514oveq1d 7431 . 2 (𝜑 → ((𝑋 𝑋) + (𝑌 𝑍)) = ((0g𝐺) + (𝑌 𝑍)))
165, 7grpsubcl 18980 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1711, 3, 4, 16syl3anc 1368 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
185, 6, 12grplid 18928 . . 3 ((𝐺 ∈ Grp ∧ (𝑌 𝑍) ∈ 𝐵) → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
1911, 17, 18syl2anc 582 . 2 (𝜑 → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
209, 15, 193eqtrd 2769 1 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6543  (class class class)co 7416  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18894  -gcsg 18896  Abelcabl 19740
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 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-minusg 18898  df-sbg 18899  df-cmn 19741  df-abl 19742
This theorem is referenced by:  hdmaprnlem7N  41384
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