Theorem shmulclt 5124

Description: Closure of vector scalar multiplication in a subspace of a Hilbert space.

Assertion

Ref	Expression
shmulclt	|- (H e. SH -> ((A e. CC /\ B e. H) -> (A .s B) e. H))

Proof of Theorem shmulclt

Step	Hyp	Ref	Expression
1		sh 5116	. . 3 |- (H e. SH <-> ((H (_ H~ /\ 0v e. H) /\ (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H)))
2	1	pm3.27bd 263	. 2 |- (H e. SH -> (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H))
3		pm3.27 260	. 2 |- ((A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H) -> A.x e. CC A.y e. H (x .s y) e. H)
4		opreq1 3006	. . . 4 |- (x = A -> (x .s y) = (A .s y))
5	4	eleq1d 1155	. . 3 |- (x = A -> ((x .s y) e. H <-> (A .s y) e. H))
6		opreq2 3007	. . . 4 |- (y = B -> (A .s y) = (A .s B))
7	6	eleq1d 1155	. . 3 |- (y = B -> ((A .s y) e. H <-> (A .s B) e. H))
8	5, 7	rcla42v 1404	. 2 |- (A.x e. CC A.y e. H (x .s y) e. H -> ((A e. CC /\ B e. H) -> (A .s B) e. H))
9	2, 3, 8	3syl 21	1 |- (H e. SH -> ((A e. CC /\ B e. H) -> (A .s B) e. H))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487  (class class class)co 3001  CCcc 4026  H~chil 4958   +v cva 4959   .s csm 4960  0vc0v 4961  SHcsh 4967

This theorem is referenced by:  shsubclt 5125  projlem18 5210  pjthlem12 5236  shscl 5282  shintcl 5294  h1de2b 5459  h1de2ctlem 5460  spansn 5462  spansnmul 5469  spansnsst 5476  spanunsn 5482  h1datom 5483  pjmul 5568  strlem1 5691

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-sh 5114

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