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Theorem grpoeqdivid 34011
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 27726 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
543adant2 1126 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
6 oveq1 6821 . . . 4 (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
76eqeq1d 2762 . . 3 (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
85, 7syl5ibrcom 237 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9 oveq1 6821 . . 3 ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
101, 2grponpcan 27727 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
111, 3grpolid 27700 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
12113adant2 1126 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
1310, 12eqeq12d 2775 . . 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 1072   = wceq 1632  wcel 2139  ran crn 5267  cfv 6049  (class class class)co 6814  GrpOpcgr 27673  GIdcgi 27674   /𝑔 cgs 27676
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-pow 4992  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-pw 4304  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-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-grpo 27677  df-gid 27678  df-ginv 27679  df-gdiv 27680
This theorem is referenced by:  grpokerinj  34023  dmncan1  34206
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