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Theorem grpoinvid1 27510
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
grpoinvid1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Proof of Theorem grpoinvid1
StepHypRef Expression
1 oveq2 6698 . . . 4 ((𝑁𝐴) = 𝐵 → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
21adantl 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
5 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
63, 4, 5grporinv 27509 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
763adant3 1101 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
87adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
92, 8eqtr3d 2687 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10 oveq2 6698 . . . 4 ((𝐴𝐺𝐵) = 𝑈 → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
1110adantl 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
123, 4, 5grpolinv 27508 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
1312oveq1d 6705 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14133adant3 1101 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
153, 5grpoinvcl 27506 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
1615adantrr 753 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
17 simprl 809 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
18 simprr 811 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
1916, 17, 183jca 1261 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋))
203grpoass 27485 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2119, 20syldan 486 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
22213impb 1279 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2314, 22eqtr3d 2687 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
243, 4grpolid 27498 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
25243adant2 1100 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
2623, 25eqtr3d 2687 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
2726adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
283, 4grporid 27499 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
2915, 28syldan 486 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
30293adant3 1101 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3130adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3211, 27, 313eqtr3rd 2694 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 895 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  ran crn 5144  cfv 5926  (class class class)co 6690  GrpOpcgr 27471  GIdcgi 27472  invcgn 27473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-grpo 27475  df-gid 27476  df-ginv 27477
This theorem is referenced by:  grpoinvop  27515  rngonegmn1l  33870
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