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Theorem rrgval 19495
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 6330 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 rrgval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2823 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6330 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
6 rrgval.t . . . . . . . . . 10 · = (.r𝑅)
75, 6syl6eqr 2823 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
87oveqd 6808 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
9 fveq2 6330 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
10 rrgval.z . . . . . . . . 9 0 = (0g𝑅)
119, 10syl6eqr 2823 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
128, 11eqeq12d 2786 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) = (0g𝑟) ↔ (𝑥 · 𝑦) = 0 ))
1311eqeq2d 2781 . . . . . . 7 (𝑟 = 𝑅 → (𝑦 = (0g𝑟) ↔ 𝑦 = 0 ))
1412, 13imbi12d 333 . . . . . 6 (𝑟 = 𝑅 → (((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ((𝑥 · 𝑦) = 0𝑦 = 0 )))
154, 14raleqbidv 3301 . . . . 5 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )))
164, 15rabeqbidv 3345 . . . 4 (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
17 df-rlreg 19491 . . . 4 RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
183fvexi 6341 . . . . 5 𝐵 ∈ V
1918rabex 4946 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} ∈ V
2016, 17, 19fvmpt 6422 . . 3 (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
21 fvprc 6324 . . . 4 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22 fvprc 6324 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
233, 22syl5eq 2817 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
24 rabeq 3342 . . . . . 6 (𝐵 = ∅ → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2523, 24syl 17 . . . . 5 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
26 rab0 4102 . . . . 5 {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅
2725, 26syl6eq 2821 . . . 4 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅)
2821, 27eqtr4d 2808 . . 3 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2920, 28pm2.61i 176 . 2 (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
301, 29eqtri 2793 1 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  c0 4063  cfv 6029  (class class class)co 6791  Basecbs 16057  .rcmulr 16143  0gc0g 16301  RLRegcrlreg 19487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5992  df-fun 6031  df-fv 6037  df-ov 6794  df-rlreg 19491
This theorem is referenced by:  isrrg  19496  rrgeq0  19498  rrgss  19500
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