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Theorem grpoeqdivid 37262
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 30304 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
543adant2 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐵) = 𝑈)
6 oveq1 7412 . . . 4 (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
76eqeq1d 2728 . . 3 (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
85, 7syl5ibrcom 246 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9 oveq1 7412 . . 3 ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
101, 2grponpcan 30305 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
111, 3grpolid 30278 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
12113adant2 1128 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
1310, 12eqeq12d 2742 . . 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 1084   = wceq 1533  wcel 2098  ran crn 5670  cfv 6537  (class class class)co 7405  GrpOpcgr 30251  GIdcgi 30252   /𝑔 cgs 30254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258
This theorem is referenced by:  grpokerinj  37274  dmncan1  37457
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