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Theorem ablpnpcan 19605
Description: Cancellation law for mixed addition and subtraction. (pnpcan 11447 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 19598 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋𝐵𝑍𝐵)) → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
91, 2, 3, 2, 4, 8syl122anc 1380 . 2 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
10 ablgrp 19574 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
111, 10syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
12 eqid 2737 . . . . 5 (0g𝐺) = (0g𝐺)
135, 12, 7grpsubid 18838 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1411, 2, 13syl2anc 585 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1514oveq1d 7377 . 2 (𝜑 → ((𝑋 𝑋) + (𝑌 𝑍)) = ((0g𝐺) + (𝑌 𝑍)))
165, 7grpsubcl 18834 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1711, 3, 4, 16syl3anc 1372 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
185, 6, 12grplid 18787 . . 3 ((𝐺 ∈ Grp ∧ (𝑌 𝑍) ∈ 𝐵) → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
1911, 17, 18syl2anc 585 . 2 (𝜑 → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
209, 15, 193eqtrd 2781 1 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  0gc0g 17328  Grpcgrp 18755  -gcsg 18757  Abelcabl 19570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-cmn 19571  df-abl 19572
This theorem is referenced by:  hdmaprnlem7N  40347
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