Theorem ablpnpcan 19786

Description: Cancellation law for mixed addition and subtraction. (pnpcan 11531 analog.) (Contributed by NM, 29-May-2015.)

Hypotheses

Ref	Expression
ablsubadd.b	⊢ 𝐵 = (Base‘𝐺)
ablsubadd.p	⊢ + = (+g‘𝐺)
ablsubadd.m	⊢ − = (-g‘𝐺)
ablsubsub.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablsubsub.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ablsubsub.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ablsubsub.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
ablpnpcan.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablpnpcan.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ablpnpcan.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ablpnpcan.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
ablpnpcan	⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

Proof of Theorem ablpnpcan

Step	Hyp	Ref	Expression
1		ablsubsub.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablsubsub.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		ablsubsub.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ablsubsub.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
5		ablsubadd.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
6		ablsubadd.p	. . . 4 ⊢ + = (+g‘𝐺)
7		ablsubadd.m	. . . 4 ⊢ − = (-g‘𝐺)
8	5, 6, 7	ablsub4 19777	. . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍)))
9	1, 2, 3, 2, 4, 8	syl122anc 1376	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍)))
10		ablgrp 19752	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
11	1, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		eqid 2725	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	5, 12, 7	grpsubid 18988	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
14	11, 2, 13	syl2anc 582	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
15	14	oveq1d 7434	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍)))
16	5, 7	grpsubcl 18984	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵)
17	11, 3, 4, 16	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵)
18	5, 6, 12	grplid 18932	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
19	11, 17, 18	syl2anc 582	. 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
20	9, 15, 19	3eqtrd 2769	1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  +_gcplusg 17236  0_gc0g 17424  Grpcgrp 18898  -_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:  hdmaprnlem7N  41458