Theorem ablnnncan 20036

Description: Cancellation law for group subtraction. (nnncan 11593 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)

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)
ablsub32.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
ablnnncan	⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌))

Proof of Theorem ablnnncan

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		ablgrp 19999	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
7	4, 6	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
8		ablnncan.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		ablsub32.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10	1, 3	grpsubcl 19230	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵)
11	7, 8, 9, 10	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 5, 11, 9	ablsubsub4 20032	. 2 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍)))
13	1, 2	ablcom 20013	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑌 − 𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍)))
14	4, 11, 9, 13	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍)))
15	1, 2, 3	ablpncan3 20030	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌)
16	4, 9, 8, 15	syl12anc 850	. . . 4 ⊢ (𝜑 → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌)
17	14, 16	eqtrd 2795	. . 3 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = 𝑌)
18	17	oveq2d 7436	. 2 ⊢ (𝜑 → (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍)) = (𝑋 − 𝑌))
19	12, 18	eqtrd 2795	1 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  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 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 5326  ax-pr 5390  ax-un 7751

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 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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: (None)