Theorem sshjval 26385

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 26033	. . 3 |- ~H e. _V
2	1	elpw2 4529	. 2 |- ( A e. ~P ~H <-> A C_ ~H )
3	1	elpw2 4529	. 2 |- ( B e. ~P ~H <-> B C_ ~H )
4		uneq12 3567	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x u. y ) = ( A u. B ) )
5	4	fveq2d 5778	. . . 4 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( x u. y ) ) = ( _|_ ` ( A u. B ) ) )
6	5	fveq2d 5778	. . 3 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( _|_ ` ( x u. y ) ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
7		df-chj 26345	. . 3 |- vH = ( x e. ~P ~H , y e. ~P ~H |-> ( _|_ ` ( _|_ ` ( x u. y ) ) ) )
8		fvex 5784	. . 3 |- ( _|_ ` ( _|_ ` ( A u. B ) ) ) e. _V
9	6, 7, 8	ovmpt2a 6332	. 2 |- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
10	2, 3, 9	syl2anbr 478	1 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1399   e. wcel 1826   u. cun 3387   C_ wss 3389   ~Pcpw 3927   ` cfv 5496  (class class class)co 6196   ~Hchil 25953   _|_cort 25964   vH chj 25967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-hilex 26033

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-chj 26345

This theorem is referenced by:  shjval  26386  sshjval3  26389  sshjcl  26390  sshjval2  26446  ssjo  26482  sshhococi  26581