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Theorem spanss2 22032
Description: A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
spanss2  |-  ( A 
C_  ~H  ->  A  C_  ( span `  A )
)

Proof of Theorem spanss2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssintub 3959 . 2  |-  A  C_  |^|
{ x  e.  SH  |  A  C_  x }
2 spanval 22020 . 2  |-  ( A 
C_  ~H  ->  ( span `  A )  =  |^| { x  e.  SH  |  A  C_  x } )
31, 2syl5sseqr 3303 1  |-  ( A 
C_  ~H  ->  A  C_  ( span `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   {crab 2623    C_ wss 3228   |^|cint 3941   ` cfv 5334   ~Hchil 21607   SHcsh 21616   spancspn 21620
This theorem is referenced by:  shsupunss  22033  spanuni  22231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-i2m1 8892  ax-1ne0 8893  ax-rrecex 8896  ax-cnre 8897  ax-hilex 21687  ax-hfvadd 21688  ax-hv0cl 21691  ax-hfvmul 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-map 6859  df-nn 9834  df-hlim 21660  df-sh 21894  df-ch 21909  df-span 21996
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