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Theorem grpoinvid1 29178
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 7345 . . . 4 ((𝑁𝐴) = 𝐵 → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
21adantl 482 . . 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 29177 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
763adant3 1131 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
87adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
92, 8eqtr3d 2778 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10 oveq2 7345 . . . 4 ((𝐴𝐺𝐵) = 𝑈 → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
1110adantl 482 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
123, 4, 5grpolinv 29176 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
1312oveq1d 7352 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14133adant3 1131 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
153, 5grpoinvcl 29174 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
1615adantrr 714 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
17 simprl 768 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
18 simprr 770 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
1916, 17, 183jca 1127 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋))
203grpoass 29153 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2119, 20syldan 591 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
22213impb 1114 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2314, 22eqtr3d 2778 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
243, 4grpolid 29166 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
25243adant2 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
2623, 25eqtr3d 2778 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
2726adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
283, 4grporid 29167 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
2915, 28syldan 591 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
30293adant3 1131 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3130adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3211, 27, 313eqtr3rd 2785 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 798 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  ran crn 5621  cfv 6479  (class class class)co 7337  GrpOpcgr 29139  GIdcgi 29140  invcgn 29141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-grpo 29143  df-gid 29144  df-ginv 29145
This theorem is referenced by:  grpoinvop  29183  rngonegmn1l  36212
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