Theorem sshjval 26988

Description: Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
sshjval	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )

Proof of Theorem sshjval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 26637	. . 3 |- ~H e. _V
2	1	elpw2 4584	. 2 |- ( A e. ~P ~H <-> A C_ ~H )
3	1	elpw2 4584	. 2 |- ( B e. ~P ~H <-> B C_ ~H )
4		uneq12 3615	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x u. y ) = ( A u. B ) )
5	4	fveq2d 5881	. . . 4 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( x u. y ) ) = ( _|_ ` ( A u. B ) ) )
6	5	fveq2d 5881	. . 3 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( _|_ ` ( x u. y ) ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
7		df-chj 26948	. . 3 |- vH = ( x e. ~P ~H , y e. ~P ~H |-> ( _|_ ` ( _|_ ` ( x u. y ) ) ) )
8		fvex 5887	. . 3 |- ( _|_ ` ( _|_ ` ( A u. B ) ) ) e. _V
9	6, 7, 8	ovmpt2a 6437	. 2 |- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
10	2, 3, 9	syl2anbr 482	1 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1868   u. cun 3434   C_ wss 3436   ~Pcpw 3979   ` cfv 5597  (class class class)co 6301   ~Hchil 26557   _|_cort 26568   vH chj 26571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-hilex 26637

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-chj 26948

This theorem is referenced by:  shjval  26989  sshjval3  26992  sshjcl  26993  sshjval2  27049  ssjo  27085  sshhococi  27184