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Theorem sshjval3 22817
 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 22463 . . . . . 6
21elpw2 4332 . . . . 5
31elpw2 4332 . . . . 5
4 uniprg 3998 . . . . 5
52, 3, 4syl2anbr 467 . . . 4
65fveq2d 5699 . . 3
76fveq2d 5699 . 2
8 prssi 3922 . . . 4
92, 3, 8syl2anbr 467 . . 3
10 hsupval 22797 . . 3
119, 10syl 16 . 2
12 sshjval 22813 . 2
137, 11, 123eqtr4rd 2455 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wcel 1721   cun 3286   wss 3288  cpw 3767  cpr 3783  cuni 3983  cfv 5421  (class class class)co 6048  chil 22383  cort 22394   chj 22397   chsup 22398 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-hilex 22463 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-chj 22773  df-chsup 22774
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