Theorem grpodivval 30916

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 30915	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
5	4	oveqd 7433	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵))
6		oveq1 7423	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝑦)))
7		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
8	7	oveq2d 7432	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝐵)))
9		eqid 2765	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))
10		ovex 7449	. . . 4 ⊢ (𝐴𝐺(𝑁‘𝐵)) ∈ V
11	6, 8, 9, 10	ovmpo 7576	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵) = (𝐴𝐺(𝑁‘𝐵)))
12	5, 11	sylan9eq 2820	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))
13	12	3impb 1132	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ran crn 5664  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  GrpOpcgr 30870  invcgn 30872   /_𝑔 cgs 30873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-gdiv 30877

This theorem is used by:  grpodivinv  30917  grpoinvdiv  30918  grpodivdiv  30921  grpomuldivass  30922  grpodivid  30923  grponpcan  30924  ablodivdiv4  30935  nvmval  31023  rngosub  38614