Theorem dvrfval 20362

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)

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
dvrfval	⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥, · ,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   / (𝑥,𝑦)

Proof of Theorem dvrfval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrval.d	. 2 ⊢ / = (/r‘𝑅)
2		fveq2 6876	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2788	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6876	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6		dvrval.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7	5, 6	eqtr4di 2788	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8		fveq2 6876	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvrval.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2788	. . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11		eqidd 2736	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
12		fveq2 6876	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
13		dvrval.i	. . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅)
14	12, 13	eqtr4di 2788	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼)
15	14	fveq1d 6878	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦))
16	10, 11, 15	oveq123d 7426	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦)))
17	4, 7, 16	mpoeq123dv 7482	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
18		df-dvr 20361	. . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
19	3	fvexi 6890	. . . . 5 ⊢ 𝐵 ∈ V
20	6	fvexi 6890	. . . . 5 ⊢ 𝑈 ∈ V
21	19, 20	mpoex 8078	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V
22	17, 18, 21	fvmpt 6986	. . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
23		fvprc 6868	. . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅)
24		fvprc 6868	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
25	3, 24	eqtrid 2782	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
26	25	orcd 873	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅))
27		0mpo0 7490	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝑈 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = ∅)
28	26, 27	syl 17	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = ∅)
29	23, 28	eqtr4d 2773	. . 3 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
30	22, 29	pm2.61i 182	. 2 ⊢ (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))
31	1, 30	eqtri 2758	1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  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:  dvrval  20363  cnflddiv  21363  cnflddivOLD  21364  dvrcn  24122