Theorem sshjval 24900

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 24548	. . 3 |- ~H e. _V
2	1	elpw2 4559	. 2 |- ( A e. ~P ~H <-> A C_ ~H )
3	1	elpw2 4559	. 2 |- ( B e. ~P ~H <-> B C_ ~H )
4		uneq12 3608	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x u. y ) = ( A u. B ) )
5	4	fveq2d 5798	. . . 4 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( x u. y ) ) = ( _|_ ` ( A u. B ) ) )
6	5	fveq2d 5798	. . 3 |- ( ( x = A /\ y = B ) -> ( _|_ ` ( _|_ ` ( x u. y ) ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
7		df-chj 24860	. . 3 |- vH = ( x e. ~P ~H , y e. ~P ~H |-> ( _|_ ` ( _|_ ` ( x u. y ) ) ) )
8		fvex 5804	. . 3 |- ( _|_ ` ( _|_ ` ( A u. B ) ) ) e. _V
9	6, 7, 8	ovmpt2a 6326	. 2 |- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
10	2, 3, 9	syl2anbr 480	1 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   u. cun 3429   C_ wss 3431   ~Pcpw 3963   ` cfv 5521  (class class class)co 6195   ~Hchil 24468   _|_cort 24479   vH chj 24482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-hilex 24548

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-chj 24860

This theorem is referenced by:  shjval  24901  sshjval3  24904  sshjcl  24905  sshjval2  24961  ssjo  24997  sshhococi  25096