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Theorem p0val 17649
Description: Value of poset zero. (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
p0val.b 𝐵 = (Base‘𝐾)
p0val.g 𝐺 = (glb‘𝐾)
p0val.z 0 = (0.‘𝐾)
Assertion
Ref Expression
p0val (𝐾𝑉0 = (𝐺𝐵))

Proof of Theorem p0val
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3498 . 2 (𝐾𝑉𝐾 ∈ V)
2 p0val.z . . 3 0 = (0.‘𝐾)
3 fveq2 6659 . . . . . 6 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
4 p0val.g . . . . . 6 𝐺 = (glb‘𝐾)
53, 4syl6eqr 2877 . . . . 5 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
6 fveq2 6659 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
7 p0val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7syl6eqr 2877 . . . . 5 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
95, 8fveq12d 6666 . . . 4 (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺𝐵))
10 df-p0 17647 . . . 4 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
11 fvex 6672 . . . 4 (𝐺𝐵) ∈ V
129, 10, 11fvmpt 6757 . . 3 (𝐾 ∈ V → (0.‘𝐾) = (𝐺𝐵))
132, 12syl5eq 2871 . 2 (𝐾 ∈ V → 0 = (𝐺𝐵))
141, 13syl 17 1 (𝐾𝑉0 = (𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cfv 6344  Basecbs 16481  glbcglb 17551  0.cp0 17645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-p0 17647
This theorem is referenced by:  p0le  17651  clatp0cl  30664  xrsp0  30695  op0cl  36392  atl0cl  36511
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