Theorem shinclt 9346

Description: Closure of intersection of two subspaces.

Assertion

Ref	Expression
shinclt	|- ((A e. SH /\ B e. SH) -> (A i^i B) e. SH)

Proof of Theorem shinclt

Step	Hyp	Ref	Expression
1		ineq1 2213	. . 3 |- (A = if(A e. SH, A, H~) -> (A i^i B) = (if(A e. SH, A, H~) i^i B))
2	1	eleq1d 1543	. 2 |- (A = if(A e. SH, A, H~) -> ((A i^i B) e. SH <-> (if(A e. SH, A, H~) i^i B) e. SH))
3		ineq2 2214	. . 3 |- (B = if(B e. SH, B, H~) -> (if(A e. SH, A, H~) i^i B) = (if(A e. SH, A, H~) i^i if(B e. SH, B, H~)))
4	3	eleq1d 1543	. 2 |- (B = if(B e. SH, B, H~) -> ((if(A e. SH, A, H~) i^i B) e. SH <-> (if(A e. SH, A, H~) i^i if(B e. SH, B, H~)) e. SH))
5		helsh 9112	. . . 4 |- H~ e. SH
6	5	elimel 2398	. . 3 |- if(A e. SH, A, H~) e. SH
7	5	elimel 2398	. . 3 |- if(B e. SH, B, H~) e. SH
8	6, 7	shincl 9326	. 2 |- (if(A e. SH, A, H~) i^i if(B e. SH, B, H~)) e. SH
9	2, 4, 8	dedth2h 2391	1 |- ((A e. SH /\ B e. SH) -> (A i^i B) e. SH)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   i^i cin 2049  ifcif 2365  H~chil 8783  SHcsh 8792

This theorem is referenced by:  orthin 9365  sumdmdi 10337

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864  ax-hfvadd 8865  ax-hv0cl 8868  ax-hfvmul 8870

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-hlim 8836  df-sh 9071  df-ch 9087

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