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Theorem ablpnpcan 18537
Description: Cancellation law for mixed addition and subtraction. (pnpcan 10610 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 18530 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋𝐵𝑍𝐵)) → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
91, 2, 3, 2, 4, 8syl122anc 1499 . 2 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
10 ablgrp 18510 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
111, 10syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
12 eqid 2797 . . . . 5 (0g𝐺) = (0g𝐺)
135, 12, 7grpsubid 17812 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1411, 2, 13syl2anc 580 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1514oveq1d 6891 . 2 (𝜑 → ((𝑋 𝑋) + (𝑌 𝑍)) = ((0g𝐺) + (𝑌 𝑍)))
165, 7grpsubcl 17808 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1711, 3, 4, 16syl3anc 1491 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
185, 6, 12grplid 17765 . . 3 ((𝐺 ∈ Grp ∧ (𝑌 𝑍) ∈ 𝐵) → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
1911, 17, 18syl2anc 580 . 2 (𝜑 → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
209, 15, 193eqtrd 2835 1 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cfv 6099  (class class class)co 6876  Basecbs 16181  +gcplusg 16264  0gc0g 16412  Grpcgrp 17735  -gcsg 17737  Abelcabl 18506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-0g 16414  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-sbg 17740  df-cmn 18507  df-abl 18508
This theorem is referenced by:  hdmaprnlem7N  37868
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