Theorem projlem21 9201

Description: Part of Lemma 3.6 of [Beran] p. 100. The hypothesis lets us work with our postulated vector sequence (whose existence was shown by projlem17 9197). Here we just show the sequence value belongs to the closed subspace H. Used by projlem27 9207 projlem28 9208.

Hypothesis

Ref	Expression
projlem21.1	|- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))

Assertion

Ref	Expression
projlem21	|- (ph -> (D e. NN -> (F` D) e. H))

Distinct variable groups:   w,A   w,D   w,F   w,R

Proof of Theorem projlem21

Step	Hyp	Ref	Expression
1		ffvelrn 3820	. . 3 |- ((F:NN-->H /\ D e. NN) -> (F` D) e. H)
2		projlem21.1	. . . 4 |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))
3	2	pm3.26bi 322	. . 3 |- (ph -> F:NN-->H)
4	1, 3	sylan 450	. 2 |- ((ph /\ D e. NN) -> (F` D) e. H)
5	4	ex 373	1 |- (ph -> (D e. NN -> (F` D) e. H))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 960  A.wral 1648   class class class wbr 2624  -->wf 3184  ` cfv 3188  (class class class)co 3969  1c1 5247   + caddc 5249   - cmin 5304   / cdiv 5306  NNcn 5308   < clt 5498   -h cmv 8787  normhcno 8789

This theorem is referenced by:  projlem27 9207  projlem28 9208

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204

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