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Theorem rrgval 19206
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 6148 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 rrgval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6148 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
6 rrgval.t . . . . . . . . . 10 · = (.r𝑅)
75, 6syl6eqr 2673 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
87oveqd 6621 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
9 fveq2 6148 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
10 rrgval.z . . . . . . . . 9 0 = (0g𝑅)
119, 10syl6eqr 2673 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
128, 11eqeq12d 2636 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) = (0g𝑟) ↔ (𝑥 · 𝑦) = 0 ))
1311eqeq2d 2631 . . . . . . 7 (𝑟 = 𝑅 → (𝑦 = (0g𝑟) ↔ 𝑦 = 0 ))
1412, 13imbi12d 334 . . . . . 6 (𝑟 = 𝑅 → (((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ((𝑥 · 𝑦) = 0𝑦 = 0 )))
154, 14raleqbidv 3141 . . . . 5 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )))
164, 15rabeqbidv 3181 . . . 4 (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
17 df-rlreg 19202 . . . 4 RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
18 fvex 6158 . . . . . 6 (Base‘𝑅) ∈ V
193, 18eqeltri 2694 . . . . 5 𝐵 ∈ V
2019rabex 4773 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} ∈ V
2116, 17, 20fvmpt 6239 . . 3 (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
22 fvprc 6142 . . . 4 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
23 fvprc 6142 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
243, 23syl5eq 2667 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
25 rabeq 3179 . . . . . 6 (𝐵 = ∅ → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2624, 25syl 17 . . . . 5 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
27 rab0 3929 . . . . 5 {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅
2826, 27syl6eq 2671 . . . 4 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅)
2922, 28eqtr4d 2658 . . 3 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
3021, 29pm2.61i 176 . 2 (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
311, 30eqtri 2643 1 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  c0 3891  cfv 5847  (class class class)co 6604  Basecbs 15781  .rcmulr 15863  0gc0g 16021  RLRegcrlreg 19198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-rlreg 19202
This theorem is referenced by:  isrrg  19207  rrgeq0  19209  rrgss  19211
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