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Theorem p0val 18323
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 3466 . 2 (𝐾𝑉𝐾 ∈ V)
2 p0val.z . . 3 0 = (0.‘𝐾)
3 fveq2 6847 . . . . . 6 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
4 p0val.g . . . . . 6 𝐺 = (glb‘𝐾)
53, 4eqtr4di 2795 . . . . 5 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
6 fveq2 6847 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
7 p0val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7eqtr4di 2795 . . . . 5 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
95, 8fveq12d 6854 . . . 4 (𝑝 = 𝐾 → ((glb‘𝑝)‘(Base‘𝑝)) = (𝐺𝐵))
10 df-p0 18321 . . . 4 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
11 fvex 6860 . . . 4 (𝐺𝐵) ∈ V
129, 10, 11fvmpt 6953 . . 3 (𝐾 ∈ V → (0.‘𝐾) = (𝐺𝐵))
132, 12eqtrid 2789 . 2 (𝐾 ∈ V → 0 = (𝐺𝐵))
141, 13syl 17 1 (𝐾𝑉0 = (𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3448  cfv 6501  Basecbs 17090  glbcglb 18206  0.cp0 18319
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-p0 18321
This theorem is referenced by:  p0le  18325  clatp0cl  31878  xrsp0  31914  op0cl  37675  atl0cl  37794
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