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Theorem clatp0cl 30833
Description: The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
Hypotheses
Ref Expression
clatp0cl.b 𝐵 = (Base‘𝑊)
clatp0cl.0 0 = (0.‘𝑊)
Assertion
Ref Expression
clatp0cl (𝑊 ∈ CLat → 0𝐵)

Proof of Theorem clatp0cl
StepHypRef Expression
1 clatp0cl.b . . 3 𝐵 = (Base‘𝑊)
2 eqid 2738 . . 3 (glb‘𝑊) = (glb‘𝑊)
3 clatp0cl.0 . . 3 0 = (0.‘𝑊)
41, 2, 3p0val 17769 . 2 (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵))
5 ssid 3899 . . 3 𝐵𝐵
61, 2clatglbcl 17842 . . 3 ((𝑊 ∈ CLat ∧ 𝐵𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
75, 6mpan2 691 . 2 (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
84, 7eqeltrd 2833 1 (𝑊 ∈ CLat → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wss 3843  cfv 6339  Basecbs 16588  glbcglb 17671  0.cp0 17765  CLatccla 17835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-lub 17702  df-glb 17703  df-p0 17767  df-clat 17836
This theorem is referenced by: (None)
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