Theorem ablnncan 20043

Description: Cancellation law for group subtraction. (nncan 11594 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 2760	. . 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 20040	. 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌))
8		ablgrp 20008	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9	4, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
10		eqid 2760	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
11	1, 10, 3	grpsubid 19243	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
12	9, 5, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
13	12	oveq1d 7439	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌))
14	1, 2, 10	grplid 19187	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌)
15	9, 6, 14	syl2anc 596	. 2 ⊢ (𝜑 → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌)
16	7, 13, 15	3eqtrd 2799	1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  0_gc0g 17619  Grpcgrp 19153  -_gcsg 19155  Abelcabl 20004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-0g 17621  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-minusg 19157  df-sbg 19158  df-cmn 20005  df-abl 20006

This theorem is used by:  ablnnncan1  20046  pgpfac1lem3  20302  rngqiprngfulem4  21619  tsmsxplem1  24481  baerlem5blem2  42906