Theorem grpodivval 31143

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 31142	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))))
5	4	oveqd 7438	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵))
6		oveq1 7428	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝑦)))
7		fveq2 6886	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
8	7	oveq2d 7437	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝐵)))
9		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))
10		ovex 7454	. . . 4 ⊢ (𝐴𝐺(𝑁‘𝐵)) ∈ V
11	6, 8, 9, 10	ovmpo 7581	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵) = (𝐴𝐺(𝑁‘𝐵)))
12	5, 11	sylan9eq 2815	. 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 1569   ∈ wcel 2144  ran crn 5651  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  GrpOpcgr 31097  invcgn 31099   /_𝑔 cgs 31100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-gdiv 31104

This theorem is used by:  grpodivinv  31144  grpoinvdiv  31145  grpodivdiv  31148  grpomuldivass  31149  grpodivid  31150  grponpcan  31151  ablodivdiv4  31162  nvmval  31250  rngosub  38958