Theorem ablpnpcan 19849

Description: Cancellation law for mixed addition and subtraction. (pnpcan 11463 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 19840	. . 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 1397	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍)))
10		ablgrp 19815	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
11	1, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		eqid 2761	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	5, 12, 7	grpsubid 19056	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
14	11, 2, 13	syl2anc 593	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
15	14	oveq1d 7405	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍)))
16	5, 7	grpsubcl 19052	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵)
17	11, 3, 4, 16	syl3anc 1389	. . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵)
18	5, 6, 12	grplid 18999	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
19	11, 17, 18	syl2anc 593	. 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍))
20	9, 15, 19	3eqtrd 2800	1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  +_gcplusg 17276  0_gc0g 17458  Grpcgrp 18965  -_gcsg 18967  Abelcabl 19811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-0g 17460  df-mgm 18664  df-sgrp 18743  df-mnd 18759  df-grp 18968  df-minusg 18969  df-sbg 18970  df-cmn 19812  df-abl 19813

This theorem is referenced by:  hdmaprnlem7N  42439