Theorem nmofval 9764

Description: The operator norm function.

Hypotheses

Ref	Expression
nmofval.1	|- X = (BaseSet` U)
nmofval.2	|- Y = (BaseSet` W)
nmofval.3	|- L = (norm` U)
nmofval.4	|- M = (norm` W)
nmofval.6	|- N = (UnormOpW)

Assertion

Ref	Expression
nmofval	|- ((U e. NrmCVec /\ W e. NrmCVec) -> N = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))})

Distinct variable groups:   y,t,L   t,M,y   x,t,z,y,U   t,W,x,y,z   t,X,x,y,z   t,Y,x,y

Proof of Theorem nmofval

Step	Hyp	Ref	Expression
1		nmofval.2	. . . . . . . 8 |- Y = (BaseSet` W)
2		fvex 4689	. . . . . . . 8 |- (BaseSet` W) e. _V
3	1, 2	eqeltri 1967	. . . . . . 7 |- Y e. _V
4		nmofval.1	. . . . . . . 8 |- X = (BaseSet` U)
5		fvex 4689	. . . . . . . 8 |- (BaseSet` U) e. _V
6	4, 5	eqeltri 1967	. . . . . . 7 |- X e. _V
7	3, 6	elmap 5393	. . . . . 6 |- (t e. (Y ^m X) <-> t:X-->Y)
8	7	anbi1i 539	. . . . 5 |- ((t e. (Y ^m X) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < )) <-> (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < )))
9	8	opabbii 3402	. . . 4 |- {<.t, y>. | (t e. (Y ^m X) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))} = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))}
10		oprex 4907	. . . . 5 |- (Y ^m X) e. _V
11	10	opabex2 4539	. . . 4 |- {<.t, y>. | (t e. (Y ^m X) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))} e. _V
12	9, 11	eqeltrri 1968	. . 3 |- {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))} e. _V
13		fveq2 4681	. . . . . . 7 |- (u = U -> (BaseSet` u) = (BaseSet` U))
14	13, 4	syl6eqr 1946	. . . . . 6 |- (u = U -> (BaseSet` u) = X)
15	14	feq2d 4557	. . . . 5 |- (u = U -> (t:(BaseSet` u)-->(BaseSet` w) <-> t:X-->(BaseSet` w)))
16		fveq2 4681	. . . . . . . . . . . . 13 |- (u = U -> (norm` u) = (norm` U))
17		nmofval.3	. . . . . . . . . . . . 13 |- L = (norm` U)
18	16, 17	syl6eqr 1946	. . . . . . . . . . . 12 |- (u = U -> (norm` u) = L)
19	18	fveq1d 4683	. . . . . . . . . . 11 |- (u = U -> ((norm` u)` z) = (L` z))
20	19	breq1d 3348	. . . . . . . . . 10 |- (u = U -> (((norm` u)` z) <_ 1 <-> (L` z) <_ 1))
21	20	anbi1d 679	. . . . . . . . 9 |- (u = U -> ((((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z))) <-> ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))))
22	14, 21	rexeqbidv 2275	. . . . . . . 8 |- (u = U -> (E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z))) <-> E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))))
23	22	abbidv 2008	. . . . . . 7 |- (u = U -> {x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))} = {x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))})
24		supeq1 5665	. . . . . . 7 |- ({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))} = {x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))} -> sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))
25	23, 24	syl 12	. . . . . 6 |- (u = U -> sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))
26	25	eqeq2d 1895	. . . . 5 |- (u = U -> (y = sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) <-> y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < )))
27	15, 26	anbi12d 690	. . . 4 |- (u = U -> ((t:(BaseSet` u)-->(BaseSet` w) /\ y = sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < )) <-> (t:X-->(BaseSet` w) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))))
28	27	opabbidv 3401	. . 3 |- (u = U -> {<.t, y>. | (t:(BaseSet` u)-->(BaseSet` w) /\ y = sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))} = {<.t, y>. | (t:X-->(BaseSet` w) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))})
29		fveq2 4681	. . . . . . 7 |- (w = W -> (BaseSet` w) = (BaseSet` W))
30	29, 1	syl6eqr 1946	. . . . . 6 |- (w = W -> (BaseSet` w) = Y)
31		feq3 4553	. . . . . 6 |- ((BaseSet` w) = Y -> (t:X-->(BaseSet` w) <-> t:X-->Y))
32	30, 31	syl 12	. . . . 5 |- (w = W -> (t:X-->(BaseSet` w) <-> t:X-->Y))
33		fveq2 4681	. . . . . . . . . . . . 13 |- (w = W -> (norm` w) = (norm` W))
34		nmofval.4	. . . . . . . . . . . . 13 |- M = (norm` W)
35	33, 34	syl6eqr 1946	. . . . . . . . . . . 12 |- (w = W -> (norm` w) = M)
36	35	fveq1d 4683	. . . . . . . . . . 11 |- (w = W -> ((norm` w)` (t` z)) = (M` (t` z)))
37	36	eqeq2d 1895	. . . . . . . . . 10 |- (w = W -> (x = ((norm` w)` (t` z)) <-> x = (M` (t` z))))
38	37	anbi2d 678	. . . . . . . . 9 |- (w = W -> (((L` z) <_ 1 /\ x = ((norm` w)` (t` z))) <-> ((L` z) <_ 1 /\ x = (M` (t` z)))))
39	38	rexbidv 2124	. . . . . . . 8 |- (w = W -> (E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z))) <-> E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))))
40	39	abbidv 2008	. . . . . . 7 |- (w = W -> {x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))} = {x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))})
41		supeq1 5665	. . . . . . 7 |- ({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))} = {x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))} -> sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))
42	40, 41	syl 12	. . . . . 6 |- (w = W -> sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))
43	42	eqeq2d 1895	. . . . 5 |- (w = W -> (y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ) <-> y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < )))
44	32, 43	anbi12d 690	. . . 4 |- (w = W -> ((t:X-->(BaseSet` w) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < )) <-> (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))))
45	44	opabbidv 3401	. . 3 |- (w = W -> {<.t, y>. | (t:X-->(BaseSet` w) /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))} = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))})
46		df-nmo 9745	. . 3 |- normOp = {<.<.u, w>., n>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ n = {<.t, y>. | (t:(BaseSet` u)-->(BaseSet` w) /\ y = sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))})}
47	12, 28, 45, 46	oprabval2 4957	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (UnormOpW) = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))})
48		nmofval.6	. 2 |- N = (UnormOpW)
49	47, 48	syl5eq 1940	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> N = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  supcsup 5663  1c1 6387   <_ cle 6448  RR*cxr 6652   < clt 6653  NrmCVeccnv 9535  BaseSetcba 9537  normcnm 9541  normOpcnmo 9741

This theorem is referenced by:  nmoval 9765  hhnmoi 11464

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-sup 5664  df-nmo 9745

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