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Theorem iscss 16912
 Description: The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssval.o
cssval.c
Assertion
Ref Expression
iscss

Proof of Theorem iscss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cssval.o . . . 4
2 cssval.c . . . 4
31, 2cssval 16911 . . 3
43eleq2d 2505 . 2
5 id 21 . . . 4
6 fvex 5744 . . . 4
75, 6syl6eqel 2526 . . 3
8 id 21 . . . 4
9 fveq2 5730 . . . . 5
109fveq2d 5734 . . . 4
118, 10eqeq12d 2452 . . 3
127, 11elab3 3091 . 2
134, 12syl6bb 254 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wceq 1653   wcel 1726  cab 2424  cvv 2958  cfv 5456  cocv 16889  ccss 16890 This theorem is referenced by:  cssi  16913  iscss2  16915  obslbs  16959  hlhillcs  32821 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-ocv 16892  df-css 16893
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