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Theorem grpoeqdivid 33279
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 27236 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
543adant2 1078 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
6 oveq1 6612 . . . 4 (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
76eqeq1d 2628 . . 3 (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
85, 7syl5ibrcom 237 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9 oveq1 6612 . . 3 ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
101, 2grponpcan 27237 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
111, 3grpolid 27210 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
12113adant2 1078 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
1310, 12eqeq12d 2641 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵))
149, 13syl5ib 234 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 𝑈𝐴 = 𝐵))
158, 14impbid 202 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1992  ran crn 5080  cfv 5850  (class class class)co 6605  GrpOpcgr 27183  GIdcgi 27184   /𝑔 cgs 27186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-grpo 27187  df-gid 27188  df-ginv 27189  df-gdiv 27190
This theorem is referenced by:  grpokerinj  33291  dmncan1  33474
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