Theorem dvrfval 20428

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 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvrval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2798	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6		dvrval.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
7	5, 6	eqtr4di 2798	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8		fveq2 6920	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvrval.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2798	. . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11		eqidd 2741	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
12		fveq2 6920	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
13		dvrval.i	. . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅)
14	12, 13	eqtr4di 2798	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼)
15	14	fveq1d 6922	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦))
16	10, 11, 15	oveq123d 7469	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦)))
17	4, 7, 16	mpoeq123dv 7525	. . . 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 20427	. . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
19	3	fvexi 6934	. . . . 5 ⊢ 𝐵 ∈ V
20	6	fvexi 6934	. . . . 5 ⊢ 𝑈 ∈ V
21	19, 20	mpoex 8120	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V
22	17, 18, 21	fvmpt 7029	. . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
23		fvprc 6912	. . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅)
24		fvprc 6912	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
25	3, 24	eqtrid 2792	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
26	25	orcd 872	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅))
27		0mpo0 7533	. . . . 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 2783	. . 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 2768	1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Basecbs 17258  ._rcmulr 17312  Unitcui 20381  inv_rcinvr 20413  /_rcdvr 20426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-dvr 20427

This theorem is referenced by:  dvrval  20429  cnflddiv  21436  cnflddivOLD  21437  dvrcn  24213