Theorem rrgval 14553

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 14552	. . 3 ⊢ (𝑧 ∈ 𝐸 → 𝑅 ∈ V)
3		elrabi 2979	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} → 𝑧 ∈ 𝐵)
4		rrgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5	4	basmex 13395	. . . 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 14549	. . . . . 6 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
8		fveq2 5693	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9	8, 4	eqtr4di 2289	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
10		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
11		rrgval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
12	10, 11	eqtr4di 2289	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
13	12	oveqd 6096	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
14		fveq2 5693	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
15		rrgval.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
16	14, 15	eqtr4di 2289	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
17	13, 16	eqeq12d 2253	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
18	16	eqeq2d 2250	. . . . . . . . 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 2765	. . . . . . 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 2816	. . . . . 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 13394	. . . . . . . . 9 ⊢ Base Fn V
24		funfvex 5710	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
25	24	funfni 5481	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
26	23, 25	mpan 428	. . . . . . . 8 ⊢ (𝑅 ∈ V → (Base‘𝑅) ∈ V)
27	4, 26	eqeltrid 2325	. . . . . . 7 ⊢ (𝑅 ∈ V → 𝐵 ∈ V)
28		rabexg 4277	. . . . . . 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 5796	. . . . 5 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
31	1, 30	eqtrid 2283	. . . 4 ⊢ (𝑅 ∈ V → 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
32	31	eleq2d 2308	. . 3 ⊢ (𝑅 ∈ V → (𝑧 ∈ 𝐸 ↔ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}))
33	2, 6, 32	pm5.21nii 716	. 2 ⊢ (𝑧 ∈ 𝐸 ↔ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
34	33	eqriv 2235	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  ._rcmulr 13415  0_gc0g 13593  RLRegcrlreg 14546

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-rlreg 14549

This theorem is referenced by:  isrrg  14554  rrgeq0  14556  rrgss  14558