Theorem dvrfvald 14146

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 14145	. . 3 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
2		fveq2 5639	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		fveq2 5639	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4		fveq2 5639	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5		eqidd 2232	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
6		fveq2 5639	. . . . . 6 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
7	6	fveq1d 5641	. . . . 5 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = ((invr‘𝑅)‘𝑦))
8	4, 5, 7	oveq123d 6038	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
9	2, 3, 8	mpoeq123dv 6082	. . 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 2816	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 13140	. . . . 5 ⊢ Base Fn V
13		funfvex 5656	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5432	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		eqidd 2232	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
17		eqidd 2232	. . . . . 6 ⊢ (𝜑 → (Unit‘𝑅) = (Unit‘𝑅))
18	16, 17, 10	unitssd 14122	. . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
19	15, 18	ssexd 4229	. . . 4 ⊢ (𝜑 → (Unit‘𝑅) ∈ V)
20		mpoexga 6376	. . . 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 5740	. 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 2232	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
28		dvrfvald.i	. . . . 5 ⊢ (𝜑 → 𝐼 = (invr‘𝑅))
29	28	fveq1d 5641	. . . 4 ⊢ (𝜑 → (𝐼‘𝑦) = ((invr‘𝑅)‘𝑦))
30	26, 27, 29	oveq123d 6038	. . 3 ⊢ (𝜑 → (𝑥 · (𝐼‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
31	24, 25, 30	mpoeq123dv 6082	. 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 2274	1 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  Basecbs 13081  ._rcmulr 13160  SRingcsrg 13975  Unitcui 14099  inv_rcinvr 14133  /_rcdvr 14144

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mgp 13933  df-srg 13976  df-dvdsr 14101  df-unit 14102  df-dvr 14145

This theorem is referenced by:  dvrvald  14147