Theorem grpasscan1 18638

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)

Hypotheses

Ref	Expression
grplcan.b	⊢ 𝐵 = (Base‘𝐺)
grplcan.p	⊢ + = (+g‘𝐺)
grpasscan1.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpasscan1	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌)

Proof of Theorem grpasscan1

Step	Hyp	Ref	Expression
1		grplcan.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grplcan.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		eqid 2738	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		grpasscan1.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
5	1, 2, 3, 4	grprinv 18629	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
6	5	3adant3 1131	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
7	6	oveq1d 7290	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = ((0g‘𝐺) + 𝑌))
8	1, 4	grpinvcl 18627	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
9	1, 2	grpass 18586	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
10	9	3exp2 1353	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))))
11	10	imp 407	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))))
12	8, 11	mpd 15	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))
13	12	3impia 1116	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
14	1, 2, 3	grplid 18609	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
15	14	3adant2 1130	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
16	7, 13, 15	3eqtr3d 2786	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  Grpcgrp 18577  inv_gcminusg 18578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581

This theorem is referenced by:  mulgaddcomlem  18726  grplsmid  31592  nsgqusf1olem3  31600