Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  sshjval3 Unicode version

Theorem sshjval3 21763
 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 21409 . . . . . 6
21elpw2 4064 . . . . 5
31elpw2 4064 . . . . 5
4 uniprg 3742 . . . . 5
52, 3, 4syl2anbr 468 . . . 4
65fveq2d 5381 . . 3
76fveq2d 5381 . 2
8 prssi 3671 . . . 4
92, 3, 8syl2anbr 468 . . 3
10 hsupval 21743 . . 3
119, 10syl 17 . 2
12 sshjval 21759 . 2
137, 11, 123eqtr4rd 2296 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1619   wcel 1621   cun 3076   wss 3078  cpw 3530  cpr 3545  cuni 3727  cfv 4592  (class class class)co 5710  chil 21329  cort 21340   chj 21343   chsup 21344 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-hilex 21409 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-chj 21719  df-chsup 21720
 Copyright terms: Public domain W3C validator