Theorem grpinvid1 18979

Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinv.p	⊢ + = (+g‘𝐺)
grpinv.u	⊢ 0 = (0g‘𝐺)
grpinv.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvid1	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))

Proof of Theorem grpinvid1

Step	Hyp	Ref	Expression
1		oveq2 7418	. . . 4 ⊢ ((𝑁‘𝑋) = 𝑌 → (𝑋 + (𝑁‘𝑋)) = (𝑋 + 𝑌))
2	1	adantl 481	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + (𝑁‘𝑋)) = (𝑋 + 𝑌))
3		grpinv.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		grpinv.p	. . . . . 6 ⊢ + = (+g‘𝐺)
5		grpinv.u	. . . . . 6 ⊢ 0 = (0g‘𝐺)
6		grpinv.n	. . . . . 6 ⊢ 𝑁 = (invg‘𝐺)
7	3, 4, 5, 6	grprinv 18978	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
8	7	3adant3 1132	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
9	8	adantr 480	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + (𝑁‘𝑋)) = 0 )
10	2, 9	eqtr3d 2773	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11		oveq2 7418	. . . 4 ⊢ ((𝑋 + 𝑌) = 0 → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = ((𝑁‘𝑋) + 0 ))
12	11	adantl 481	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = ((𝑁‘𝑋) + 0 ))
13	3, 4, 5, 6	grplinv 18977	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
14	13	oveq1d 7425	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
15	14	3adant3 1132	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
16	3, 6	grpinvcl 18975	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
17	16	adantrr 717	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁‘𝑋) ∈ 𝐵)
18		simprl 770	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
19		simprr 772	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
20	17, 18, 19	3jca 1128	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁‘𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
21	3, 4	grpass 18930	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ((𝑁‘𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
22	20, 21	syldan 591	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
23	22	3impb 1114	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
24	15, 23	eqtr3d 2773	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
25	3, 4, 5	grplid 18955	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ( 0 + 𝑌) = 𝑌)
26	25	3adant2 1131	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + 𝑌) = 𝑌)
27	24, 26	eqtr3d 2773	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = 𝑌)
28	27	adantr 480	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = 𝑌)
29	3, 4, 5	grprid 18956	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
30	16, 29	syldan 591	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
31	30	3adant3 1132	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
32	31	adantr 480	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
33	12, 28, 32	3eqtr3rd 2780	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘𝑋) = 𝑌)
34	10, 33	impbida 800	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  0_gc0g 17458  Grpcgrp 18921  inv_gcminusg 18922

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925

This theorem is referenced by:  grpinvid  18987  grpinvcnv  18994  grpinvadd  19006  subginv  19121  qusinv  19178  ghminv  19211  symginv  19388  frgpinv  19750  cnaddinv  19857  rngmneg1  20132  ringnegl  20267  lmodindp1  20976  lmodvsinv2  21000  cnfldneg  21363  zringinvg  21431  mdetunilem6  22560  invrvald  22619  dchrinv  27229  elrgspnlem1  33242  baerlem3lem1  41731