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Theorem domnrrg 20571
Description: In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
isdomn2.b 𝐵 = (Base‘𝑅)
isdomn2.t 𝐸 = (RLReg‘𝑅)
isdomn2.z 0 = (0g𝑅)
Assertion
Ref Expression
domnrrg ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐸)

Proof of Theorem domnrrg
StepHypRef Expression
1 isdomn2.b . . . . 5 𝐵 = (Base‘𝑅)
2 isdomn2.t . . . . 5 𝐸 = (RLReg‘𝑅)
3 isdomn2.z . . . . 5 0 = (0g𝑅)
41, 2, 3isdomn2 20570 . . . 4 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
54simprbi 497 . . 3 (𝑅 ∈ Domn → (𝐵 ∖ { 0 }) ⊆ 𝐸)
653ad2ant1 1132 . 2 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → (𝐵 ∖ { 0 }) ⊆ 𝐸)
7 simp2 1136 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐵)
8 simp3 1137 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋0 )
9 eldifsn 4720 . . 3 (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋𝐵𝑋0 ))
107, 8, 9sylanbrc 583 . 2 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋 ∈ (𝐵 ∖ { 0 }))
116, 10sseldd 3922 1 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  wne 2943  cdif 3884  wss 3887  {csn 4561  cfv 6433  Basecbs 16912  0gc0g 17150  NzRingcnzr 20528  RLRegcrlreg 20550  Domncdomn 20551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-rlreg 20554  df-domn 20555
This theorem is referenced by:  deg1ldgdomn  25259
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