Theorem grpoeqdivid 38084

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 30621	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈)
5	4	3adant2 1132	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈)
6		oveq1 7367	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵))
7	6	eqeq1d 2739	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈))
8	5, 7	syl5ibrcom 247	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈))
9		oveq1 7367	. . 3 ⊢ ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵))
10	1, 2	grponpcan 30622	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
11	1, 3	grpolid 30595	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
12	11	3adant2 1132	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
13	10, 12	eqeq12d 2753	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵))
14	9, 13	imbitrid 244	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 𝑈 → 𝐴 = 𝐵))
15	8, 14	impbid 212	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ran crn 5626  ‘cfv 6493  (class class class)co 7360  GrpOpcgr 30568  GIdcgi 30569   /_𝑔 cgs 30571

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-grpo 30572  df-gid 30573  df-ginv 30574  df-gdiv 30575

This theorem is referenced by:  grpokerinj  38096  dmncan1  38279