Theorem dvrval 20312

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 7394	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦)))
2		fveq2 6858	. . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌))
3	2	oveq2d 7403	. 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 20311	. 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))
10		ovex 7420	. 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V
11	1, 3, 9, 10	ovmpo 7549	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  ._rcmulr 17221  Unitcui 20264  inv_rcinvr 20296  /_rcdvr 20309

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-dvr 20310

This theorem is referenced by:  dvrcl  20313  unitdvcl  20314  dvrid  20315  dvr1  20316  dvrass  20317  dvrcan1  20318  dvrdir  20321  rdivmuldivd  20322  ringinvdv  20323  subrgdv  20498  abvdiv  20738  cnflddiv  21312  cnflddivOLD  21313  nmdvr  24558  sum2dchr  27185  dvrcan5  33187