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Theorem issh2 21618
 Description: Subspace of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh2
Distinct variable group:   ,,

Proof of Theorem issh2
StepHypRef Expression
1 issh 21617 . 2
2 ax-hfvadd 21410 . . . . . . 7
3 ffun 5248 . . . . . . 7
42, 3ax-mp 10 . . . . . 6
5 xpss12 4699 . . . . . . . 8
65anidms 629 . . . . . . 7
72fdmi 5251 . . . . . . 7
86, 7syl6sseqr 3146 . . . . . 6
9 funimassov 5849 . . . . . 6
104, 8, 9sylancr 647 . . . . 5
11 ax-hfvmul 21415 . . . . . . 7
12 ffun 5248 . . . . . . 7
1311, 12ax-mp 10 . . . . . 6
14 xpss2 4703 . . . . . . 7
1511fdmi 5251 . . . . . . 7
1614, 15syl6sseqr 3146 . . . . . 6
17 funimassov 5849 . . . . . 6
1813, 16, 17sylancr 647 . . . . 5
1910, 18anbi12d 694 . . . 4
2019adantr 453 . . 3
2120pm5.32i 621 . 2
221, 21bitri 242 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wcel 1621  wral 2509   wss 3078   cxp 4578   cdm 4580  cima 4583   wfun 4586  wf 4588  (class class class)co 5710  cc 8615  chil 21329   cva 21330   csm 21331  c0v 21334  csh 21338 This theorem is referenced by:  shaddcl  21626  shmulcl  21627  shmulclOLD  21628  issh3  21629  helch  21653  hsn0elch  21657  hhshsslem2  21675  ocsh  21692  shscli  21726  shintcli  21738  imaelshi  22468 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-hilex 21409  ax-hfvadd 21410  ax-hfvmul 21415 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-ov 5713  df-sh 21616
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