Theorem ablnncan 19699

Description: Cancellation law for group subtraction. (nncan 11393 analog.) (Contributed by NM, 7-Apr-2015.)

Hypotheses

Ref	Expression
ablnncan.b	⊢ 𝐵 = (Base‘𝐺)
ablnncan.m	⊢ − = (-g‘𝐺)
ablnncan.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablnncan.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ablnncan.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
ablnncan	⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)

Proof of Theorem ablnncan

Step	Hyp	Ref	Expression
1		ablnncan.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2729	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablnncan.m	. . 3 ⊢ − = (-g‘𝐺)
4		ablnncan.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
5		ablnncan.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		ablnncan.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	1, 2, 3, 4, 5, 5, 6	ablsubsub 19696	. 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌))
8		ablgrp 19664	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9	4, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
10		eqid 2729	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
11	1, 10, 3	grpsubid 18903	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
12	9, 5, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
13	12	oveq1d 7364	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌))
14	1, 2, 10	grplid 18846	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌)
15	9, 6, 14	syl2anc 584	. 2 ⊢ (𝜑 → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌)
16	7, 13, 15	3eqtrd 2768	1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343  Grpcgrp 18812  -_gcsg 18814  Abelcabl 19660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-minusg 18816  df-sbg 18817  df-cmn 19661  df-abl 19662

This theorem is referenced by:  ablnnncan1  19702  pgpfac1lem3  19958  rngqiprngfulem4  21221  tsmsxplem1  24038  baerlem5blem2  41691