Theorem dvrval 20319

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
dvrval.b	⊢ 𝐵 = (Base‘𝑅)
dvrval.t	⊢ · = (.r‘𝑅)
dvrval.u	⊢ 𝑈 = (Unit‘𝑅)
dvrval.i	⊢ 𝐼 = (invr‘𝑅)
dvrval.d	⊢ / = (/r‘𝑅)

Assertion

Ref	Expression
dvrval	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌)))

Proof of Theorem dvrval

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

Step	Hyp	Ref	Expression
1		oveq1 7353	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦)))
2		fveq2 6822	. . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌))
3	2	oveq2d 7362	. 2 ⊢ (𝑦 = 𝑌 → (𝑋 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑌)))
4		dvrval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
5		dvrval.t	. . 3 ⊢ · = (.r‘𝑅)
6		dvrval.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
7		dvrval.i	. . 3 ⊢ 𝐼 = (invr‘𝑅)
8		dvrval.d	. . 3 ⊢ / = (/r‘𝑅)
9	4, 5, 6, 7, 8	dvrfval 20318	. 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))
10		ovex 7379	. 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V
11	1, 3, 9, 10	ovmpo 7506	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  ._rcmulr 17159  Unitcui 20271  inv_rcinvr 20303  /_rcdvr 20316

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-dvr 20317

This theorem is referenced by:  dvrcl  20320  unitdvcl  20321  dvrid  20322  dvr1  20323  dvrass  20324  dvrcan1  20325  dvrdir  20328  rdivmuldivd  20329  ringinvdv  20330  subrgdv  20502  abvdiv  20742  cnflddiv  21335  cnflddivOLD  21336  nmdvr  24583  sum2dchr  27210  dvrcan5  33198