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Theorem rrgeq0i 21243
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 21242 . . 3 (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 )))
65simprbi 495 . 2 (𝑋 ∈ 𝐸 β†’ βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 ))
7 oveq2 7434 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
87eqeq1d 2730 . . . 4 (𝑦 = π‘Œ β†’ ((𝑋 Β· 𝑦) = 0 ↔ (𝑋 Β· π‘Œ) = 0 ))
9 eqeq1 2732 . . . 4 (𝑦 = π‘Œ β†’ (𝑦 = 0 ↔ π‘Œ = 0 ))
108, 9imbi12d 343 . . 3 (𝑦 = π‘Œ β†’ (((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 ) ↔ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 )))
1110rspcv 3607 . 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 505 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 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  .rcmulr 17241  0gc0g 17428  RLRegcrlreg 21233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-rlreg 21237
This theorem is referenced by:  rrgeq0  21244  znrrg  21506  deg1mul2  26070  rrgsubm  32976  rlocf1  33012  fracerl  33017
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