Theorem grpinvid1 18675

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 7315	. . . 4 ⊢ ((𝑁‘𝑋) = 𝑌 → (𝑋 + (𝑁‘𝑋)) = (𝑋 + 𝑌))
2	1	adantl 483	. . 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 18674	. . . . 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 482	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + (𝑁‘𝑋)) = 0 )
10	2, 9	eqtr3d 2778	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11		oveq2 7315	. . . 4 ⊢ ((𝑋 + 𝑌) = 0 → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = ((𝑁‘𝑋) + 0 ))
12	11	adantl 483	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = ((𝑁‘𝑋) + 0 ))
13	3, 4, 5, 6	grplinv 18673	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
14	13	oveq1d 7322	. . . . . . 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 18672	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
17	16	adantrr 715	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁‘𝑋) ∈ 𝐵)
18		simprl 769	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
19		simprr 771	. . . . . . . . 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 18631	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ((𝑁‘𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
22	20, 21	syldan 592	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
23	22	3impb 1115	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑁‘𝑋) + 𝑋) + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
24	15, 23	eqtr3d 2778	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + 𝑌) = ((𝑁‘𝑋) + (𝑋 + 𝑌)))
25	3, 4, 5	grplid 18654	. . . . . 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 2778	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = 𝑌)
28	27	adantr 482	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + (𝑋 + 𝑌)) = 𝑌)
29	3, 4, 5	grprid 18655	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
30	16, 29	syldan 592	. . . . 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 482	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁‘𝑋) + 0 ) = (𝑁‘𝑋))
33	12, 28, 32	3eqtr3rd 2785	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘𝑋) = 𝑌)
34	10, 33	impbida 799	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ‘cfv 6458  (class class class)co 7307  Basecbs 16957  +_gcplusg 17007  0_gc0g 17195  Grpcgrp 18622  inv_gcminusg 18623

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-riota 7264  df-ov 7310  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-grp 18625  df-minusg 18626

This theorem is referenced by:  grpinvid  18681  grpinvcnv  18688  grpinvadd  18698  subginv  18807  qusinv  18860  ghminv  18886  symginv  19055  frgpinv  19415  cnaddinv  19517  ringnegl  19878  lmodindp1  20321  lmodvsinv2  20344  cnfldneg  20669  zringinvg  20732  mdetunilem6  21811  invrvald  21870  dchrinv  26454  baerlem3lem1  39763