Theorem grpinvid2 19103

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
grpinvid2	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))

Proof of Theorem grpinvid2

Step	Hyp	Ref	Expression
1		oveq1 7426	. . . 4 ⊢ ((𝑁‘𝑋) = 𝑌 → ((𝑁‘𝑋) + 𝑋) = (𝑌 + 𝑋))
2	1	adantl 487	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → ((𝑁‘𝑋) + 𝑋) = (𝑌 + 𝑋))
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	grplinv 19100	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
8	7	3adant3 1150	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
9	8	adantr 486	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → ((𝑁‘𝑋) + 𝑋) = 0 )
10	2, 9	eqtr3d 2802	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
11	3, 6	grpinvcl 19098	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
12	3, 4, 5	grplid 19078	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
13	11, 12	syldan 603	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
14	13	3adant3 1150	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
15	14	eqcomd 2771	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 + (𝑁‘𝑋)))
16	15	adantr 486	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁‘𝑋) = ( 0 + (𝑁‘𝑋)))
17		oveq1 7426	. . . 4 ⊢ ((𝑌 + 𝑋) = 0 → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = ( 0 + (𝑁‘𝑋)))
18	17	adantl 487	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = ( 0 + (𝑁‘𝑋)))
19		simprr 785	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
20		simprl 783	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
21	11	adantrr 730	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁‘𝑋) ∈ 𝐵)
22	19, 20, 21	3jca 1146	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵))
23	3, 4	grpass 19053	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
24	22, 23	syldan 603	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
25	24	3impb 1132	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
26	3, 4, 5, 6	grprinv 19101	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
27	26	oveq2d 7435	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑌 + (𝑋 + (𝑁‘𝑋))) = (𝑌 + 0 ))
28	27	3adant3 1150	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 + (𝑋 + (𝑁‘𝑋))) = (𝑌 + 0 ))
29	3, 4, 5	grprid 19079	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + 0 ) = 𝑌)
30	29	3adant2 1149	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 + 0 ) = 𝑌)
31	25, 28, 30	3eqtrd 2804	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = 𝑌)
32	31	adantr 486	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = 𝑌)
33	16, 18, 32	3eqtr2d 2806	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁‘𝑋) = 𝑌)
34	10, 33	impbida 813	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  0_gc0g 17514  Grpcgrp 19044  inv_gcminusg 19045

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7376  df-ov 7422  df-0g 17516  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-grp 19047  df-minusg 19048

This theorem is used by:  grpinvcnv  19117  grpsubeq0  19136  prdsinvgd  19161  xpsinv  19170  eqg0subg  19311  rngmneg2  20290  ringnegr  20432  islindf4  22038  psrneg  22158  pi1inv  25262  fldhmf1  42915  lindslinindimp2lem4  49298  lincresunit3  49318