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Theorem rrgeq0i 19990
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 19989 . . 3 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
65simprbi 497 . 2 (𝑋𝐸 → ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ))
7 oveq2 7153 . . . . 5 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
87eqeq1d 2820 . . . 4 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9 eqeq1 2822 . . . 4 (𝑦 = 𝑌 → (𝑦 = 0𝑌 = 0 ))
108, 9imbi12d 346 . . 3 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0𝑌 = 0 )))
1110rspcv 3615 . 2 (𝑌𝐵 → (∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ) → ((𝑋 · 𝑌) = 0𝑌 = 0 )))
126, 11mpan9 507 1 ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  0gc0g 16701  RLRegcrlreg 19980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-rlreg 19984
This theorem is referenced by:  rrgeq0  19991  znrrg  20640  deg1mul2  24635
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