Theorem grpodivid 30523

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 2735	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
3		grpdivf.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivval 30516	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
5	4	3anidm23 1423	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
6		grpdivid.3	. . 3 ⊢ 𝑈 = (GId‘𝐺)
7	1, 6, 2	grporinv 30508	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = 𝑈)
8	5, 7	eqtrd 2770	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ran crn 5655  ‘cfv 6531  (class class class)co 7405  GrpOpcgr 30470  GIdcgi 30471  invcgn 30472   /_𝑔 cgs 30473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-grpo 30474  df-gid 30475  df-ginv 30476  df-gdiv 30477

This theorem is referenced by:  ablonncan  30537  grpoeqdivid  37905