Theorem grpinvid1 18870

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 7357	. . . 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 18869	. . . . 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 2766	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11		oveq2 7357	. . . 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 18868	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
14	13	oveq1d 7364	. . . . . . 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 18866	. . . . . . . . . 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 18821	. . . . . . . 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 2766	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
25	3, 4, 5	grplid 18846	. . . . . 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 2766	. . . 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 18847	. . . . . 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 2773	. 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 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343  Grpcgrp 18812  inv_gcminusg 18813

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-riota 7306  df-ov 7352  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-minusg 18816

This theorem is referenced by:  grpinvid  18878  grpinvcnv  18885  grpinvadd  18897  subginv  19012  qusinv  19069  ghminv  19102  symginv  19281  frgpinv  19643  cnaddinv  19750  rngmneg1  20052  ringnegl  20187  lmodindp1  20917  lmodvsinv2  20941  cnfldneg  21302  zringinvg  21372  mdetunilem6  22502  invrvald  22561  dchrinv  27170  elrgspnlem1  33182  baerlem3lem1  41690