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Theorem shmulclOLD 22713
 Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
shmulclOLD

Proof of Theorem shmulclOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issh2 22703 . . . 4
21simprbi 451 . . 3
32simprd 450 . 2
4 oveq1 6080 . . . 4
54eleq1d 2501 . . 3
6 oveq2 6081 . . . 4
76eleq1d 2501 . . 3
85, 7rspc2v 3050 . 2
93, 8syl5com 28 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697   wss 3312  (class class class)co 6073  cc 8980  chil 22414   cva 22415   csm 22416  c0v 22419  csh 22423 This theorem is referenced by:  mayete3iOLD  23223 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494  ax-hfvadd 22495  ax-hfvmul 22500 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-sh 22701
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