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Theorem rrgeq0i 19686
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 19685 . . 3 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
65simprbi 492 . 2 (𝑋𝐸 → ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ))
7 oveq2 6930 . . . . 5 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
87eqeq1d 2779 . . . 4 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9 eqeq1 2781 . . . 4 (𝑦 = 𝑌 → (𝑦 = 0𝑌 = 0 ))
108, 9imbi12d 336 . . 3 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0𝑌 = 0 )))
1110rspcv 3506 . 2 (𝑌𝐵 → (∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ) → ((𝑋 · 𝑌) = 0𝑌 = 0 )))
126, 11mpan9 502 1 ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  wral 3089  cfv 6135  (class class class)co 6922  Basecbs 16255  .rcmulr 16339  0gc0g 16486  RLRegcrlreg 19676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-rlreg 19680
This theorem is referenced by:  rrgeq0  19687  znrrg  20309  deg1mul2  24311
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