Theorem rrgval 13742

Description: Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgval.e	⊢ 𝐸 = (RLReg‘𝑅)
rrgval.b	⊢ 𝐵 = (Base‘𝑅)
rrgval.t	⊢ · = (.r‘𝑅)
rrgval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
rrgval	⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   · (𝑥,𝑦)   𝐸(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem rrgval

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. . . 4 ⊢ 𝐸 = (RLReg‘𝑅)
2	1	rrgmex 13741	. . 3 ⊢ (𝑧 ∈ 𝐸 → 𝑅 ∈ V)
3		elrabi 2913	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} → 𝑧 ∈ 𝐵)
4		rrgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5	4	basmex 12667	. . . 4 ⊢ (𝑧 ∈ 𝐵 → 𝑅 ∈ V)
6	3, 5	syl 14	. . 3 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} → 𝑅 ∈ V)
7		df-rlreg 13738	. . . . . 6 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
8		fveq2 5546	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9	8, 4	eqtr4di 2244	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
10		fveq2 5546	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
11		rrgval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
12	10, 11	eqtr4di 2244	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
13	12	oveqd 5927	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
14		fveq2 5546	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
15		rrgval.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
16	14, 15	eqtr4di 2244	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
17	13, 16	eqeq12d 2208	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
18	16	eqeq2d 2205	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑦 = (0g‘𝑟) ↔ 𝑦 = 0 ))
19	17, 18	imbi12d 234	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
20	9, 19	raleqbidv 2706	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
21	9, 20	rabeqbidv 2755	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
22		id 19	. . . . . 6 ⊢ (𝑅 ∈ V → 𝑅 ∈ V)
23		basfn 12666	. . . . . . . . 9 ⊢ Base Fn V
24		funfvex 5563	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
25	24	funfni 5346	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
26	23, 25	mpan 424	. . . . . . . 8 ⊢ (𝑅 ∈ V → (Base‘𝑅) ∈ V)
27	4, 26	eqeltrid 2280	. . . . . . 7 ⊢ (𝑅 ∈ V → 𝐵 ∈ V)
28		rabexg 4172	. . . . . . 7 ⊢ (𝐵 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈ V)
29	27, 28	syl 14	. . . . . 6 ⊢ (𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈ V)
30	7, 21, 22, 29	fvmptd3 5643	. . . . 5 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
31	1, 30	eqtrid 2238	. . . 4 ⊢ (𝑅 ∈ V → 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
32	31	eleq2d 2263	. . 3 ⊢ (𝑅 ∈ V → (𝑧 ∈ 𝐸 ↔ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}))
33	2, 6, 32	pm5.21nii 705	. 2 ⊢ (𝑧 ∈ 𝐸 ↔ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
34	33	eqriv 2190	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   Fn wfn 5241  ‘cfv 5246  (class class class)co 5910  Basecbs 12608  ._rcmulr 12686  0_gc0g 12857  RLRegcrlreg 13735

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-ov 5913  df-inn 8973  df-ndx 12611  df-slot 12612  df-base 12614  df-rlreg 13738

This theorem is referenced by:  isrrg  13743  rrgeq0  13745  rrgss  13746