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Theorem pltfval 18271
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 3493 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6881 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4eqtr4di 2791 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 4119 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 18270 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
84fvexi 6895 . . . . 5 ∈ V
98difexi 5324 . . . 4 ( ∖ I ) ∈ V
106, 7, 9fvmpt 6987 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
112, 10syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
121, 11eqtrid 2785 1 (𝐾𝐴< = ( ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3943   I cid 5569  cfv 6535  lecple 17191  ltcplt 18248
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 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-plt 18270
This theorem is referenced by:  pltval  18272  relt  21141  opsrtoslem2  21585  oppglt  32103  xrslt  32148  submarchi  32303
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