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Theorem rrgval 19988
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 · = (.r𝑅)
rrgval.z 0 = (0g𝑅)
Assertion
Ref Expression
rrgval 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 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 variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . 2 𝐸 = (RLReg‘𝑅)
2 fveq2 6663 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 rrgval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2871 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6663 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
6 rrgval.t . . . . . . . . . 10 · = (.r𝑅)
75, 6syl6eqr 2871 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
87oveqd 7162 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
9 fveq2 6663 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
10 rrgval.z . . . . . . . . 9 0 = (0g𝑅)
119, 10syl6eqr 2871 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
128, 11eqeq12d 2834 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) = (0g𝑟) ↔ (𝑥 · 𝑦) = 0 ))
1311eqeq2d 2829 . . . . . . 7 (𝑟 = 𝑅 → (𝑦 = (0g𝑟) ↔ 𝑦 = 0 ))
1412, 13imbi12d 346 . . . . . 6 (𝑟 = 𝑅 → (((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ((𝑥 · 𝑦) = 0𝑦 = 0 )))
154, 14raleqbidv 3399 . . . . 5 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )))
164, 15rabeqbidv 3483 . . . 4 (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
17 df-rlreg 19984 . . . 4 RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
183fvexi 6677 . . . . 5 𝐵 ∈ V
1918rabex 5226 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} ∈ V
2016, 17, 19fvmpt 6761 . . 3 (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
21 fvprc 6656 . . . 4 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22 fvprc 6656 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
233, 22syl5eq 2865 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2423rabeqdv 3482 . . . . 5 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
25 rab0 4334 . . . . 5 {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅
2624, 25syl6eq 2869 . . . 4 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅)
2721, 26eqtr4d 2856 . . 3 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2820, 27pm2.61i 183 . 2 (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
291, 28eqtri 2841 1 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  c0 4288  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  0gc0g 16701  RLRegcrlreg 19980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-rlreg 19984
This theorem is referenced by:  isrrg  19989  rrgeq0  19991  rrgss  19993
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