Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  p1val Structured version   Visualization version   GIF version

Theorem p1val 17711
 Description: Value of poset zero. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
p1val.b 𝐵 = (Base‘𝐾)
p1val.u 𝑈 = (lub‘𝐾)
p1val.t 1 = (1.‘𝐾)
Assertion
Ref Expression
p1val (𝐾𝑉1 = (𝑈𝐵))

Proof of Theorem p1val
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3429 . 2 (𝐾𝑉𝐾 ∈ V)
2 p1val.t . . 3 1 = (1.‘𝐾)
3 fveq2 6659 . . . . . 6 (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾))
4 p1val.u . . . . . 6 𝑈 = (lub‘𝐾)
53, 4eqtr4di 2812 . . . . 5 (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈)
6 fveq2 6659 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
7 p1val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7eqtr4di 2812 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
95, 8fveq12d 6666 . . . 4 (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈𝐵))
10 df-p1 17709 . . . 4 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘)))
11 fvex 6672 . . . 4 (𝑈𝐵) ∈ V
129, 10, 11fvmpt 6760 . . 3 (𝐾 ∈ V → (1.‘𝐾) = (𝑈𝐵))
132, 12syl5eq 2806 . 2 (𝐾 ∈ V → 1 = (𝑈𝐵))
141, 13syl 17 1 (𝐾𝑉1 = (𝑈𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ‘cfv 6336  Basecbs 16534  lubclub 17611  1.cp1 17707 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-p1 17709 This theorem is referenced by:  ple1  17713  clatp1cl  30774  xrsp1  30810  op1cl  36754
 Copyright terms: Public domain W3C validator