Theorem vccl 8194

Description: Closure of the scalar product of a complex vector space.

Hypotheses

Ref	Expression
vci.1	|- G = (1st` W)
vci.2	|- S = (2nd` W)
vci.3	|- X = ran G

Assertion

Ref	Expression
vccl	|- ((W e. CVec /\ A e. CC /\ B e. X) -> (ASB) e. X)

Proof of Theorem vccl

Step	Hyp	Ref	Expression
1		foprrn 4051	. 2 |- ((S:(CC X. X)-->X /\ A e. CC /\ B e. X) -> (ASB) e. X)
2		vci.1	. . 3 |- G = (1st` W)
3		vci.2	. . 3 |- S = (2nd` W)
4		vci.3	. . 3 |- X = ran G
5	2, 3, 4	vcsm 8193	. 2 |- (W e. CVec -> S:(CC X. X)-->X)
6	1, 5	syl3an1 863	1 |- ((W e. CVec /\ A e. CC /\ B e. X) -> (ASB) e. X)

Syntax hints:   -> wi 3   /\ w3a 779   = wceq 960   e. wcel 962   X. cxp 3184  ran crn 3187  -->wf 3194  ` cfv 3198  (class class class)co 3979  1stc1st 4093  2ndc2nd 4094  CCcc 5252  CVeccvc 8189

This theorem is referenced by:  vc0 8213  vcm 8215  vcnegsubdi2 8219  vcsub4 8220  nvscl 8272

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-fv 3214  df-opr 3981  df-1st 4095  df-2nd 4096  df-vc 8190

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