Theorem ablpnpcan 19681

Description: Cancellation law for mixed addition and subtraction. (pnpcan 11495 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 19672	. . 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 1379	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍)))
10		ablgrp 19647	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
11	1, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		eqid 2732	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	5, 12, 7	grpsubid 18903	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
14	11, 2, 13	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
15	14	oveq1d 7420	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍)))
16	5, 7	grpsubcl 18899	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵)
17	11, 3, 4, 16	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵)
18	5, 6, 12	grplid 18848	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
19	11, 17, 18	syl2anc 584	. 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
20	9, 15, 19	3eqtrd 2776	1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Grpcgrp 18815  -_gcsg 18817  Abelcabl 19643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-cmn 19644  df-abl 19645

This theorem is referenced by:  hdmaprnlem7N  40714