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

Theorem pltfval 16880
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 3198 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6148 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4syl6eqr 2673 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 3705 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 16879 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8 fvex 6158 . . . . . 6 (le‘𝐾) ∈ V
94, 8eqeltri 2694 . . . . 5 ∈ V
10 difexg 4768 . . . . 5 ( ∈ V → ( ∖ I ) ∈ V)
119, 10ax-mp 5 . . . 4 ( ∖ I ) ∈ V
126, 7, 11fvmpt 6239 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
132, 12syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
141, 13syl5eq 2667 1 (𝐾𝐴< = ( ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552   I cid 4984  cfv 5847  lecple 15869  ltcplt 16862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-plt 16879
This theorem is referenced by:  pltval  16881  opsrtoslem2  19404  relt  19880  oppglt  29439  xrslt  29461  submarchi  29525
  Copyright terms: Public domain W3C validator