Theorem rrgval 13828

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 13827	. . 3 ⊢ (𝑧 ∈ 𝐸 → 𝑅 ∈ V)
3		elrabi 2917	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} → 𝑧 ∈ 𝐵)
4		rrgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5	4	basmex 12747	. . . 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 13824	. . . . . 6 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
8		fveq2 5559	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9	8, 4	eqtr4di 2247	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
10		fveq2 5559	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
11		rrgval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
12	10, 11	eqtr4di 2247	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
13	12	oveqd 5940	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
14		fveq2 5559	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
15		rrgval.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
16	14, 15	eqtr4di 2247	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
17	13, 16	eqeq12d 2211	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
18	16	eqeq2d 2208	. . . . . . . . 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 2709	. . . . . . 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 2758	. . . . . 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 12746	. . . . . . . . 9 ⊢ Base Fn V
24		funfvex 5576	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
25	24	funfni 5359	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
26	23, 25	mpan 424	. . . . . . . 8 ⊢ (𝑅 ∈ V → (Base‘𝑅) ∈ V)
27	4, 26	eqeltrid 2283	. . . . . . 7 ⊢ (𝑅 ∈ V → 𝐵 ∈ V)
28		rabexg 4177	. . . . . . 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 5656	. . . . 5 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
31	1, 30	eqtrid 2241	. . . 4 ⊢ (𝑅 ∈ V → 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
32	31	eleq2d 2266	. . 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 2193	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479  Vcvv 2763   Fn wfn 5254  ‘cfv 5259  (class class class)co 5923  Basecbs 12688  ._rcmulr 12766  0_gc0g 12937  RLRegcrlreg 13821

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7972  ax-resscn 7973  ax-1re 7975  ax-addrcl 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5926  df-inn 8993  df-ndx 12691  df-slot 12692  df-base 12694  df-rlreg 13824

This theorem is referenced by:  isrrg  13829  rrgeq0  13831  rrgss  13832