Theorem grpodivid 29192

Description: Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdivf.1	⊢ 𝑋 = ran 𝐺
grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)
grpdivid.3	⊢ 𝑈 = (GId‘𝐺)

Assertion

Ref	Expression
grpodivid	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈)

Proof of Theorem grpodivid

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		eqid 2736	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
3		grpdivf.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivval 29185	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
5	4	3anidm23 1420	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
6		grpdivid.3	. . 3 ⊢ 𝑈 = (GId‘𝐺)
7	1, 6, 2	grporinv 29177	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = 𝑈)
8	5, 7	eqtrd 2776	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ran crn 5621  ‘cfv 6479  (class class class)co 7337  GrpOpcgr 29139  GIdcgi 29140  invcgn 29141   /_𝑔 cgs 29142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-grpo 29143  df-gid 29144  df-ginv 29145  df-gdiv 29146

This theorem is referenced by:  ablonncan  29206  grpoeqdivid  36144