Theorem ablnncan 20034

Description: Cancellation law for group subtraction. (nncan 11587 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 2761	. . 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 20031	. 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌))
8		ablgrp 19999	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9	4, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
10		eqid 2761	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
11	1, 10, 3	grpsubid 19234	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺))
12	9, 5, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺))
13	12	oveq1d 7435	. 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌))
14	1, 2, 10	grplid 19178	. . 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 2800	1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Grpcgrp 19144  -_gcsg 19146  Abelcabl 19995

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-cmn 19996  df-abl 19997

This theorem is used by:  ablnnncan1  20037  pgpfac1lem3  20293  rngqiprngfulem4  21610  tsmsxplem1  24472  baerlem5blem2  42769