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Theorem rrgeq0i 13744
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 13743 . . 3 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
65simprbi 275 . 2 (𝑋𝐸 → ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ))
7 oveq2 5918 . . . . 5 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
87eqeq1d 2202 . . . 4 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9 eqeq1 2200 . . . 4 (𝑦 = 𝑌 → (𝑦 = 0𝑌 = 0 ))
108, 9imbi12d 234 . . 3 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0𝑌 = 0 )))
1110rspcv 2860 . 2 (𝑌𝐵 → (∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 ) → ((𝑋 · 𝑌) = 0𝑌 = 0 )))
126, 11mpan9 281 1 ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  wral 2472  cfv 5246  (class class class)co 5910  Basecbs 12608  .rcmulr 12686  0gc0g 12857  RLRegcrlreg 13735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-ov 5913  df-inn 8973  df-ndx 12611  df-slot 12612  df-base 12614  df-rlreg 13738
This theorem is referenced by:  rrgeq0  13745  znrrg  14125
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