Theorem dvrfvald 14137

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
dvrfvald.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
dvrfvald.t	⊢ (𝜑 → · = (.r‘𝑅))
dvrfvald.u	⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
dvrfvald.i	⊢ (𝜑 → 𝐼 = (invr‘𝑅))
dvrfvald.d	⊢ (𝜑 → / = (/r‘𝑅))
dvrfvald.r	⊢ (𝜑 → 𝑅 ∈ SRing)

Assertion

Ref	Expression
dvrfvald	⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

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

Allowed substitution hints:   / (𝑥,𝑦)

Proof of Theorem dvrfvald

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvr 14136	. . 3 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
2		fveq2 5635	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		fveq2 5635	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4		fveq2 5635	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5		eqidd 2230	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
6		fveq2 5635	. . . . . 6 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
7	6	fveq1d 5637	. . . . 5 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = ((invr‘𝑅)‘𝑦))
8	4, 5, 7	oveq123d 6034	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
9	2, 3, 8	mpoeq123dv 6078	. . 3 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
10		dvrfvald.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing)
11	10	elexd 2814	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 13131	. . . . 5 ⊢ Base Fn V
13		funfvex 5652	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5429	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
17		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (Unit‘𝑅) = (Unit‘𝑅))
18	16, 17, 10	unitssd 14113	. . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
19	15, 18	ssexd 4227	. . . 4 ⊢ (𝜑 → (Unit‘𝑅) ∈ V)
20		mpoexga 6372	. . . 4 ⊢ (((Base‘𝑅) ∈ V ∧ (Unit‘𝑅) ∈ V) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ V)
21	15, 19, 20	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ V)
22	1, 9, 11, 21	fvmptd3 5736	. 2 ⊢ (𝜑 → (/r‘𝑅) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
23		dvrfvald.d	. 2 ⊢ (𝜑 → / = (/r‘𝑅))
24		dvrfvald.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
25		dvrfvald.u	. . 3 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
26		dvrfvald.t	. . . 4 ⊢ (𝜑 → · = (.r‘𝑅))
27		eqidd 2230	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
28		dvrfvald.i	. . . . 5 ⊢ (𝜑 → 𝐼 = (invr‘𝑅))
29	28	fveq1d 5637	. . . 4 ⊢ (𝜑 → (𝐼‘𝑦) = ((invr‘𝑅)‘𝑦))
30	26, 27, 29	oveq123d 6034	. . 3 ⊢ (𝜑 → (𝑥 · (𝐼‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
31	24, 25, 30	mpoeq123dv 6078	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
32	22, 23, 31	3eqtr4d 2272	1 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013   ∈ cmpo 6015  Basecbs 13072  ._rcmulr 13151  SRingcsrg 13966  Unitcui 14090  inv_rcinvr 14124  /_rcdvr 14135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mgp 13924  df-srg 13967  df-dvdsr 14092  df-unit 14093  df-dvr 14136

This theorem is referenced by:  dvrvald  14138