Theorem sshjval3 28213

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 CH. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
sshjval3	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( \/H ` { A , B } ) )

Proof of Theorem sshjval3

Step	Hyp	Ref	Expression
1		ax-hilex 27856	. . . . . 6 |- ~H e. _V
2	1	elpw2 4828	. . . . 5 |- ( A e. ~P ~H <-> A C_ ~H )
3	1	elpw2 4828	. . . . 5 |- ( B e. ~P ~H <-> B C_ ~H )
4		uniprg 4450	. . . . 5 |- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> U. { A , B } = ( A u. B ) )
5	2, 3, 4	syl2anbr 497	. . . 4 |- ( ( A C_ ~H /\ B C_ ~H ) -> U. { A , B } = ( A u. B ) )
6	5	fveq2d 6195	. . 3 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( _|_ ` U. { A , B } ) = ( _|_ ` ( A u. B ) ) )
7	6	fveq2d 6195	. 2 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( _|_ ` ( _|_ ` U. { A , B } ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
8		prssi 4353	. . . 4 |- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> { A , B } C_ ~P ~H )
9	2, 3, 8	syl2anbr 497	. . 3 |- ( ( A C_ ~H /\ B C_ ~H ) -> { A , B } C_ ~P ~H )
10		hsupval 28193	. . 3 |- ( { A , B } C_ ~P ~H -> ( \/H ` { A , B } ) = ( _|_ ` ( _|_ ` U. { A , B } ) ) )
11	9, 10	syl 17	. 2 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( \/H ` { A , B } ) = ( _|_ ` ( _|_ ` U. { A , B } ) ) )
12		sshjval 28209	. 2 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
13	7, 11, 12	3eqtr4rd 2667	1 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( \/H ` { A , B } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {cpr 4179   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   _|_cort 27787   vH chj 27790   \/H chsup 27791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-chj 28169  df-chsup 28170

This theorem is referenced by: (None)