Theorem grpoeqdivid 37387

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 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grpoeqdivid	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))

Proof of Theorem grpoeqdivid

Step	Hyp	Ref	Expression
1		grpeqdivid.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpeqdivid.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
3		grpeqdivid.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐺)
4	1, 2, 3	grpodivid 30372	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈)
5	4	3adant2 1128	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈)
6		oveq1 7433	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
7	6	eqeq1d 2730	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
8	5, 7	syl5ibrcom 246	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9		oveq1 7433	. . 3 ⊢ ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
10	1, 2	grponpcan 30373	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
11	1, 3	grpolid 30346	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
12	11	3adant2 1128	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
13	10, 12	eqeq12d 2744	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵))
14	9, 13	imbitrid 243	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 𝑈 → 𝐴 = 𝐵))
15	8, 14	impbid 211	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5683  ‘cfv 6553  (class class class)co 7426  GrpOpcgr 30319  GIdcgi 30320   /_𝑔 cgs 30322

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-grpo 30323  df-gid 30324  df-ginv 30325  df-gdiv 30326

This theorem is referenced by:  grpokerinj  37399  dmncan1  37582