Theorem htthlem4 8619

Description: Lemma for htthi 8628. Value of a functional F` k.

Hypotheses

Ref	Expression
htthlem3.1	|- X = (Base` U)
htthlem3.p	|- P = (.i` U)
htthlem3.l	|- L = (U LnOp U)
htthlem3.b	|- B = (U BLnOp U)
htthlem3.u	|- U e. CHil
htthlem3.t	|- T e. L
htthlem3.a	|- ((x e. X /\ y e. X) -> ((T` x)Py) = (xP(T` y)))
htthlem3.f	|- F = {<.m, w>. | (m e. NN /\ w = {<.v, u>. | (v e. X /\ u = ((T` v)P(f` m)))})}
htthlem3.c	|- C = <.<. + , x. >., abs>.
htthlem3.d	|- D = (U BLnOp C)
htthlem3.n	|- N = (norm` U)
htthlem3.o	|- O = (UnormOpC)

Assertion

Ref	Expression
htthlem4	|- ((A e. X /\ k e. NN) -> ((F` k)` A) = ((T` A)P(f` k)))

Distinct variable groups:   v,u,x,y,A   C,k   D,k   k,F   f,k,N   u,m,v,w,x,y,P   u,k,v,x,y   f,m,u,v,w,x,y,T,k   U,k,u,v   f,X,k,m,u,v,w,x,y

Proof of Theorem htthlem4

Step	Hyp	Ref	Expression
1		htthlem3.1	. . . 4 |- X = (Base` U)
2		htthlem3.p	. . . 4 |- P = (.i` U)
3		htthlem3.l	. . . 4 |- L = (U LnOp U)
4		htthlem3.b	. . . 4 |- B = (U BLnOp U)
5		htthlem3.u	. . . 4 |- U e. CHil
6		htthlem3.t	. . . 4 |- T e. L
7		htthlem3.a	. . . 4 |- ((x e. X /\ y e. X) -> ((T` x)Py) = (xP(T` y)))
8		htthlem3.f	. . . 4 |- F = {<.m, w>. | (m e. NN /\ w = {<.v, u>. | (v e. X /\ u = ((T` v)P(f` m)))})}
9		htthlem3.c	. . . 4 |- C = <.<. + , x. >., abs>.
10		htthlem3.d	. . . 4 |- D = (U BLnOp C)
11		htthlem3.n	. . . 4 |- N = (norm` U)
12		htthlem3.o	. . . 4 |- O = (UnormOpC)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	htthlem3 8618	. . 3 |- (k e. NN -> (F` k) = {<.v, u>. | (v e. X /\ u = ((T` v)P(f` k)))})
14	13	fveq1d 3732	. 2 |- (k e. NN -> ((F` k)` A) = ({<.v, u>. | (v e. X /\ u = ((T` v)P(f` k)))}` A))
15		fveq2 3730	. . . 4 |- (v = A -> (T` v) = (T` A))
16	15	opreq1d 3981	. . 3 |- (v = A -> ((T` v)P(f` k)) = ((T` A)P(f` k)))
17		eqid 1478	. . 3 |- {<.v, u>. | (v e. X /\ u = ((T` v)P(f` k)))} = {<.v, u>. | (v e. X /\ u = ((T` v)P(f` k)))}
18		oprex 3989	. . 3 |- ((T` A)P(f` k)) e. V
19	16, 17, 18	fvopab4 3786	. 2 |- (A e. X -> ({<.v, u>. | (v e. X /\ u = ((T` v)P(f` k)))}` A) = ((T` A)P(f` k)))
20	14, 19	sylan9eqr 1532	1 |- ((A e. X /\ k e. NN) -> ((F` k)` A) = ((T` A)P(f` k)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  <.cop 2415  {copab 2671  ` cfv 3188  (class class class)co 3969   + caddc 5249   x. cmul 5251  NNcn 5308  abscabs 6751  Basecba 8201  normcnm 8205  .icip 8345   LnOp clno 8397  normOpcnmo 8398   BLnOp cblo 8399  CHilchl 8585

This theorem is referenced by:  htthlem6 8621  htthlem9 8624

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-9 967  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-rep 2698  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-rel 3191  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-fv 3204  df-opr 3971

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