MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvid2 Structured version   Visualization version   GIF version

Theorem grpoinvid2 27713
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1 𝑋 = ran 𝐺
grpinv.2 𝑈 = (GId‘𝐺)
grpinv.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvid2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Proof of Theorem grpoinvid2
StepHypRef Expression
1 oveq1 6821 . . . 4 ((𝑁𝐴) = 𝐵 → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
21adantl 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
5 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
63, 4, 5grpolinv 27710 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
763adant3 1127 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
87adantr 472 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
92, 8eqtr3d 2796 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
103, 5grpoinvcl 27708 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
113, 4grpolid 27700 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1210, 11syldan 488 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
13123adant3 1127 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1413eqcomd 2766 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
1514adantr 472 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
16 oveq1 6821 . . . 4 ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
1716adantl 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
18 simprr 813 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
19 simprl 811 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
2010adantrr 755 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
2118, 19, 203jca 1123 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
223grpoass 27687 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
2321, 22syldan 488 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
24233impb 1108 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
253, 4, 5grporinv 27711 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
2625oveq2d 6830 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
27263adant3 1127 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
283, 4grporid 27701 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
29283adant2 1126 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
3024, 27, 293eqtrd 2798 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3130adantr 472 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3215, 17, 313eqtr2d 2800 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 913 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  ran crn 5267  cfv 6049  (class class class)co 6814  GrpOpcgr 27673  GIdcgi 27674  invcgn 27675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-grpo 27677  df-gid 27678  df-ginv 27679
This theorem is referenced by:  rngonegmn1r  34072
  Copyright terms: Public domain W3C validator