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Theorem sshjval3 26999
 Description: Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice . (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sshjval3

Proof of Theorem sshjval3
StepHypRef Expression
1 ax-hilex 26644 . . . . . 6
21elpw2 4586 . . . . 5
31elpw2 4586 . . . . 5
4 uniprg 4231 . . . . 5
52, 3, 4syl2anbr 483 . . . 4
65fveq2d 5883 . . 3
76fveq2d 5883 . 2
8 prssi 4154 . . . 4
92, 3, 8syl2anbr 483 . . 3
10 hsupval 26979 . . 3
119, 10syl 17 . 2
12 sshjval 26995 . 2
137, 11, 123eqtr4rd 2475 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1438   wcel 1869   cun 3435   wss 3437  cpw 3980  cpr 3999  cuni 4217  cfv 5599  (class class class)co 6303  chil 26564  cort 26575   chj 26578   chsup 26579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-hilex 26644 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-chj 26955  df-chsup 26956 This theorem is referenced by: (None)
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