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Theorem pltfval 17557
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 3510 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6663 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4syl6eqr 2871 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 4095 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 17556 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
84fvexi 6677 . . . . 5 ∈ V
98difexi 5223 . . . 4 ( ∖ I ) ∈ V
106, 7, 9fvmpt 6761 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
112, 10syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
121, 11syl5eq 2865 1 (𝐾𝐴< = ( ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930   I cid 5452  cfv 6348  lecple 16560  ltcplt 17539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-plt 17556
This theorem is referenced by:  pltval  17558  opsrtoslem2  20193  relt  20687  oppglt  30568  xrslt  30590  submarchi  30742
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