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Theorem rrgeq0i 20905
Description: Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e 𝐸 = (RLRegβ€˜π‘…)
rrgval.b 𝐡 = (Baseβ€˜π‘…)
rrgval.t Β· = (.rβ€˜π‘…)
rrgval.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
rrgeq0i ((𝑋 ∈ 𝐸 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 ))
Proof of Theorem rrgeq0i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . . . 4 𝐸 = (RLRegβ€˜π‘…)
2 rrgval.b . . . 4 𝐡 = (Baseβ€˜π‘…)
3 rrgval.t . . . 4 Β· = (.rβ€˜π‘…)
4 rrgval.z . . . 4 0 = (0gβ€˜π‘…)
51, 2, 3, 4isrrg 20904 . . 3 (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 )))
65simprbi 498 . 2 (𝑋 ∈ 𝐸 β†’ βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 ))
7 oveq2 7417 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
87eqeq1d 2735 . . . 4 (𝑦 = π‘Œ β†’ ((𝑋 Β· 𝑦) = 0 ↔ (𝑋 Β· π‘Œ) = 0 ))
9 eqeq1 2737 . . . 4 (𝑦 = π‘Œ β†’ (𝑦 = 0 ↔ π‘Œ = 0 ))
108, 9imbi12d 345 . . 3 (𝑦 = π‘Œ β†’ (((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 ) ↔ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 )))
1110rspcv 3609 . 2 (π‘Œ ∈ 𝐡 β†’ (βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 ) β†’ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 )))
126, 11mpan9 508 1 ((𝑋 ∈ 𝐸 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  0gc0g 17385  RLRegcrlreg 20895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-rlreg 20899
This theorem is referenced by:  rrgeq0  20906  znrrg  21121  deg1mul2  25632
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