Theorem dvrval 20363

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 7412	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦)))
2		fveq2 6876	. . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌))
3	2	oveq2d 7421	. 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 20362	. 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))
10		ovex 7438	. 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V
11	1, 3, 9, 10	ovmpo 7567	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  ._rcmulr 17272  Unitcui 20315  inv_rcinvr 20347  /_rcdvr 20360

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-dvr 20361

This theorem is referenced by:  dvrcl  20364  unitdvcl  20365  dvrid  20366  dvr1  20367  dvrass  20368  dvrcan1  20369  dvrdir  20372  rdivmuldivd  20373  ringinvdv  20374  subrgdv  20549  abvdiv  20789  cnflddiv  21363  cnflddivOLD  21364  nmdvr  24609  sum2dchr  27237  dvrcan5  33231