Theorem projlem22 9202

Description: Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 9207.

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 3730	. . . . . 6 |- (w = D -> (F` w) = (F` D))
2	1	opreq1d 3981	. . . . 5 |- (w = D -> ((F` w) -h A) = ((F` D) -h A))
3	2	fveq2d 3734	. . . 4 |- (w = D -> (normh` ((F` w) -h A)) = (normh` ((F` D) -h A)))
4		opreq2 3975	. . . . 5 |- (w = D -> (1 / w) = (1 / D))
5	4	opreq2d 3982	. . . 4 |- (w = D -> (R + (1 / w)) = (R + (1 / D)))
6	3, 5	breq12d 2636	. . 3 |- (w = D -> ((normh` ((F` w) -h A)) < (R + (1 / w)) <-> (normh` ((F` D) -h A)) < (R + (1 / D))))
7	6	rcla4v 1876	. 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 1709	. . 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 958   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

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

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-ral 1652  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-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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