Theorem grpinvid2 13699

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 6035	. . . 4 ⊢ ((𝑁‘𝑋) = 𝑌 → ((𝑁‘𝑋) + 𝑋) = (𝑌 + 𝑋))
2	1	adantl 277	. . 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 13696	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
8	7	3adant3 1044	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
9	8	adantr 276	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → ((𝑁‘𝑋) + 𝑋) = 0 )
10	2, 9	eqtr3d 2266	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑁‘𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
11	3, 6	grpinvcl 13694	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
12	3, 4, 5	grplid 13677	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
13	11, 12	syldan 282	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
14	13	3adant3 1044	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 + (𝑁‘𝑋)) = (𝑁‘𝑋))
15	14	eqcomd 2237	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 + (𝑁‘𝑋)))
16	15	adantr 276	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁‘𝑋) = ( 0 + (𝑁‘𝑋)))
17		oveq1 6035	. . . 4 ⊢ ((𝑌 + 𝑋) = 0 → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = ( 0 + (𝑁‘𝑋)))
18	17	adantl 277	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = ( 0 + (𝑁‘𝑋)))
19		simprr 533	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
20		simprl 531	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
21	11	adantrr 479	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁‘𝑋) ∈ 𝐵)
22	19, 20, 21	3jca 1204	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵))
23	3, 4	grpass 13655	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
24	22, 23	syldan 282	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
25	24	3impb 1226	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = (𝑌 + (𝑋 + (𝑁‘𝑋))))
26	3, 4, 5, 6	grprinv 13697	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
27	26	oveq2d 6044	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑌 + (𝑋 + (𝑁‘𝑋))) = (𝑌 + 0 ))
28	27	3adant3 1044	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 + (𝑋 + (𝑁‘𝑋))) = (𝑌 + 0 ))
29	3, 4, 5	grprid 13678	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + 0 ) = 𝑌)
30	29	3adant2 1043	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 + 0 ) = 𝑌)
31	25, 28, 30	3eqtrd 2268	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = 𝑌)
32	31	adantr 276	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁‘𝑋)) = 𝑌)
33	16, 18, 32	3eqtr2d 2270	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁‘𝑋) = 𝑌)
34	10, 33	impbida 600	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ‘cfv 5333  (class class class)co 6028  Basecbs 13145  +_gcplusg 13223  0_gc0g 13402  Grpcgrp 13646  inv_gcminusg 13647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650

This theorem is referenced by:  grpinvcnv  13714  grpsubeq0  13732  prdsinvgd  13756  rngmneg2  14025  ringnegr  14129  psrneg  14771