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

Theorem pltfval 18376
Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pltval.l = (le‘𝐾)
pltval.s < = (lt‘𝐾)
Assertion
Ref Expression
pltfval (𝐾𝐴< = ( ∖ I ))

Proof of Theorem pltfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pltval.s . 2 < = (lt‘𝐾)
2 elex 3501 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6906 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4eqtr4di 2795 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 4125 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 18375 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
84fvexi 6920 . . . . 5 ∈ V
98difexi 5330 . . . 4 ( ∖ I ) ∈ V
106, 7, 9fvmpt 7016 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
112, 10syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
121, 11eqtrid 2789 1 (𝐾𝐴< = ( ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948   I cid 5577  cfv 6561  lecple 17304  ltcplt 18354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-plt 18375
This theorem is referenced by:  pltval  18377  relt  21633  opsrtoslem2  22080  oppglt  32953  xrslt  33009  submarchi  33193
  Copyright terms: Public domain W3C validator