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Theorem laspwcl 14668
 Description: Closure of the supremum (join) of two lattice elements. (Contributed by NM, 12-Jun-2008.)
Hypothesis
Ref Expression
isla.1
Assertion
Ref Expression
laspwcl

Proof of Theorem laspwcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isla.1 . . . . 5
21isla 14667 . . . 4
3 simpl 445 . . . . . . 7
43ralimi 2783 . . . . . 6
54ralimi 2783 . . . . 5
65adantl 454 . . . 4
72, 6sylbi 189 . . 3
8 preq1 3885 . . . . . 6
98oveq2d 6099 . . . . 5
109eleq1d 2504 . . . 4
11 preq2 3886 . . . . . 6
1211oveq2d 6099 . . . . 5
1312eleq1d 2504 . . . 4
1410, 13rspc2v 3060 . . 3
157, 14mpan9 457 . 2
16153impb 1150 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cpr 3817   cdm 4880  (class class class)co 6083  cps 14626   cspw 14628   cinf 14629  cla 14630 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086  df-lar 14635
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