Theorem ablpnpcan 19912

Description: Cancellation law for mixed addition and subtraction. (pnpcan 11508 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 19903	. . 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 1406	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍)))
10		ablgrp 19878	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
11	1, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		eqid 2765	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	5, 12, 7	grpsubid 19113	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
14	11, 2, 13	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
15	14	oveq1d 7431	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍)))
16	5, 7	grpsubcl 19109	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵)
17	11, 3, 4, 16	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵)
18	5, 6, 12	grplid 19057	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
19	11, 17, 18	syl2anc 596	. 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
20	9, 15, 19	3eqtrd 2804	1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7416  Basecbs 17286  +_gcplusg 17327  0_gc0g 17509  Grpcgrp 19023  -_gcsg 19025  Abelcabl 19874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-0g 17511  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-grp 19026  df-minusg 19027  df-sbg 19028  df-cmn 19875  df-abl 19876

This theorem is used by:  hdmaprnlem7N  42662