Theorem projlem22 9123

Description: Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 9128.

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
projlem22	|- (ph -> (D e. NN -> (normh` ((F` D) -h A)) < (R + (1 / D))))

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

Proof of Theorem projlem22

Step	Hyp	Ref	Expression
1		fveq2 3709	. . . . . 6 |- (w = D -> (F` w) = (F` D))
2	1	opreq1d 3960	. . . . 5 |- (w = D -> ((F` w) -h A) = ((F` D) -h A))
3	2	fveq2d 3713	. . . 4 |- (w = D -> (normh` ((F` w) -h A)) = (normh` ((F` D) -h A)))
4		opreq2 3954	. . . . 5 |- (w = D -> (1 / w) = (1 / D))
5	4	opreq2d 3961	. . . 4 |- (w = D -> (R + (1 / w)) = (R + (1 / D)))
6	3, 5	breq12d 2621	. . 3 |- (w = D -> ((normh` ((F` w) -h A)) < (R + (1 / w)) <-> (normh` ((F` D) -h A)) < (R + (1 / D))))
7	6	rcla4v 1864	. 2 |- (D e. NN -> (A.w e. NN (normh` ((F` w) -h A)) < (R + (1 / w)) -> (normh` ((F` D) -h A)) < (R + (1 / D))))
8		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)))))
9	8	pm3.27bi 326	. . 3 |- (ph -> A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w))))
10		pm3.27 323	. . . 4 |- (((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w))) -> (normh` ((F` w) -h A)) < (R + (1 / w)))
11	10	r19.20si 1698	. . 3 |- (A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w))) -> A.w e. NN (normh` ((F` w) -h A)) < (R + (1 / w)))
12	9, 11	syl 10	. 2 |- (ph -> A.w e. NN (normh` ((F` w) -h A)) < (R + (1 / w)))
13	7, 12	syl5com 52	1 |- (ph -> (D e. NN -> (normh` ((F` D) -h A)) < (R + (1 / D))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   class class class wbr 2609  -->wf 3168  ` cfv 3172  (class class class)co 3948  1c1 5207   + caddc 5209   - cmin 5264   / cdiv 5266  NNcn 5268   < clt 5458   -h cmv 8731  normhcno 8733

This theorem is referenced by:  projlem27 9128

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950

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