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Theorem rrgeq0i 19270
Description: Property of a left-regular element. (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
rrgeq0i ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))

Proof of Theorem rrgeq0i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . . . 4 𝐸 = (RLReg‘𝑅)
2 rrgval.b . . . 4 𝐵 = (Base‘𝑅)
3 rrgval.t . . . 4 · = (.r𝑅)
4 rrgval.z . . . 4 0 = (0g𝑅)
51, 2, 3, 4isrrg 19269 . . 3 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
65simprbi 480 . 2 (𝑋𝐸 → ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ))
7 oveq2 6643 . . . . 5 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
87eqeq1d 2622 . . . 4 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9 eqeq1 2624 . . . 4 (𝑦 = 𝑌 → (𝑦 = 0𝑌 = 0 ))
108, 9imbi12d 334 . . 3 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0𝑌 = 0 )))
1110rspcv 3300 . 2 (𝑌𝐵 → (∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ) → ((𝑋 · 𝑌) = 0𝑌 = 0 )))
126, 11mpan9 486 1 ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  cfv 5876  (class class class)co 6635  Basecbs 15838  .rcmulr 15923  0gc0g 16081  RLRegcrlreg 19260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-rlreg 19264
This theorem is referenced by:  rrgeq0  19271  znrrg  19895  deg1mul2  23855
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