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Theorem grpoeqdivid 36340
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 29484 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
543adant2 1131 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
6 oveq1 7364 . . . 4 (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
76eqeq1d 2738 . . 3 (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
85, 7syl5ibrcom 246 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9 oveq1 7364 . . 3 ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
101, 2grponpcan 29485 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
111, 3grpolid 29458 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
12113adant2 1131 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
1310, 12eqeq12d 2752 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵))
149, 13imbitrid 243 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 𝑈𝐴 = 𝐵))
158, 14impbid 211 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  wcel 2106  ran crn 5634  cfv 6496  (class class class)co 7357  GrpOpcgr 29431  GIdcgi 29432   /𝑔 cgs 29434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-grpo 29435  df-gid 29436  df-ginv 29437  df-gdiv 29438
This theorem is referenced by:  grpokerinj  36352  dmncan1  36535
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