Theorem dvrfval 20208

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 6888	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6888	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6		dvrval.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7	5, 6	eqtr4di 2790	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8		fveq2 6888	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvrval.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2790	. . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11		eqidd 2733	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
12		fveq2 6888	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
13		dvrval.i	. . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅)
14	12, 13	eqtr4di 2790	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼)
15	14	fveq1d 6890	. . . . . 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 7480	. . . 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 20207	. . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
19	3	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
20	6	fvexi 6902	. . . . 5 ⊢ 𝑈 ∈ V
21	19, 20	mpoex 8062	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V
22	17, 18, 21	fvmpt 6995	. . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
23		fvprc 6880	. . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅)
24		fvprc 6880	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
25	3, 24	eqtrid 2784	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
26	25	orcd 871	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅))
27		0mpo0 7488	. . . . 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 2775	. . 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 2760	1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))

Syntax hints:  ¬ wn 3   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4321  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17140  ._rcmulr 17194  Unitcui 20161  inv_rcinvr 20193  /_rcdvr 20206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-dvr 20207

This theorem is referenced by:  dvrval  20209  cnflddiv  20967  dvrcn  23679