Theorem grpodivval 28003

Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdiv.1	⊢ 𝑋 = ran 𝐺
grpdiv.2	⊢ 𝑁 = (inv‘𝐺)
grpdiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grpodivval	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))

Proof of Theorem grpodivval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpdiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpdiv.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
3		grpdiv.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivfval 28002	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
5	4	oveqd 7033	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵))
6		oveq1 7023	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝑦)))
7		fveq2 6538	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
8	7	oveq2d 7032	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝐵)))
9		eqid 2795	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))
10		ovex 7048	. . . 4 ⊢ (𝐴𝐺(𝑁‘𝐵)) ∈ V
11	6, 8, 9, 10	ovmpo 7166	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵) = (𝐴𝐺(𝑁‘𝐵)))
12	5, 11	sylan9eq 2851	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))
13	12	3impb 1108	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ran crn 5444  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018  GrpOpcgr 27957  invcgn 27959   /_𝑔 cgs 27960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-gdiv 27964

This theorem is referenced by:  grpodivinv  28004  grpoinvdiv  28005  grpodivdiv  28008  grpomuldivass  28009  grpodivid  28010  grponpcan  28011  ablodivdiv4  28022  nvmval  28110  rngosub  34740