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Theorem isrrg 13759
Description: Membership in the set of left-regular elements. (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
isrrg (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Distinct variable groups:   𝑦,𝐵   𝑦,𝑅   𝑦,𝑋
Allowed substitution hints:   · (𝑦)   𝐸(𝑦)   0 (𝑦)

Proof of Theorem isrrg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 5925 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
21eqeq1d 2202 . . . 4 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
32imbi1d 231 . . 3 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0𝑦 = 0 )))
43ralbidv 2494 . 2 (𝑥 = 𝑋 → (∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
5 rrgval.e . . 3 𝐸 = (RLReg‘𝑅)
6 rrgval.b . . 3 𝐵 = (Base‘𝑅)
7 rrgval.t . . 3 · = (.r𝑅)
8 rrgval.z . . 3 0 = (0g𝑅)
95, 6, 7, 8rrgval 13758 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
104, 9elrab2 2919 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  wral 2472  cfv 5254  (class class class)co 5918  Basecbs 12618  .rcmulr 12696  0gc0g 12867  RLRegcrlreg 13751
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-rlreg 13754
This theorem is referenced by:  rrgeq0i  13760  unitrrg  13763
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