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Theorem grpoeqdivid 37868
Description: Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
grpeqdivid.1 𝑋 = ran 𝐺
grpeqdivid.2 𝑈 = (GId‘𝐺)
grpeqdivid.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpoeqdivid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))

Proof of Theorem grpoeqdivid
StepHypRef Expression
1 grpeqdivid.1 . . . . 5 𝑋 = ran 𝐺
2 grpeqdivid.3 . . . . 5 𝐷 = ( /𝑔𝐺)
3 grpeqdivid.2 . . . . 5 𝑈 = (GId‘𝐺)
41, 2, 3grpodivid 30571 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
543adant2 1130 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
6 oveq1 7438 . . . 4 (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
76eqeq1d 2737 . . 3 (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
85, 7syl5ibrcom 247 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9 oveq1 7438 . . 3 ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
101, 2grponpcan 30572 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
111, 3grpolid 30545 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
12113adant2 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
1310, 12eqeq12d 2751 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵))
149, 13imbitrid 244 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 𝑈𝐴 = 𝐵))
158, 14impbid 212 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518  GIdcgi 30519   /𝑔 cgs 30521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-gdiv 30525
This theorem is referenced by:  grpokerinj  37880  dmncan1  38063
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