Theorem nmosetre 27619

Description: The set in the supremum of the operator norm definition df-nmoo 27600 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmosetre.2	|- Y = ( BaseSet ` W )
nmosetre.4	|- N = ( normCV ` W )

Assertion

Ref	Expression
nmosetre	|- ( ( W e. NrmCVec /\ T : X --> Y ) -> { x | E. z e. X ( ( M ` z ) <_ 1 /\ x = ( N ` ( T ` z ) ) ) } C_ RR )

Distinct variable groups:   x, z, T   x, W, z   x, X, z   x, Y, z

Allowed substitution hints:   M( x, z)   N( x, z)

Proof of Theorem nmosetre

Step	Hyp	Ref	Expression
1		ffvelrn 6357	. . . . . . . . 9 |- ( ( T : X --> Y /\ z e. X ) -> ( T ` z ) e. Y )
2		nmosetre.2	. . . . . . . . . 10 |- Y = ( BaseSet ` W )
3		nmosetre.4	. . . . . . . . . 10 |- N = ( normCV ` W )
4	2, 3	nvcl 27516	. . . . . . . . 9 |- ( ( W e. NrmCVec /\ ( T ` z ) e. Y ) -> ( N ` ( T ` z ) ) e. RR )
5	1, 4	sylan2 491	. . . . . . . 8 |- ( ( W e. NrmCVec /\ ( T : X --> Y /\ z e. X ) ) -> ( N ` ( T ` z ) ) e. RR )
6	5	anassrs 680	. . . . . . 7 |- ( ( ( W e. NrmCVec /\ T : X --> Y ) /\ z e. X ) -> ( N ` ( T ` z ) ) e. RR )
7		eleq1 2689	. . . . . . 7 |- ( x = ( N ` ( T ` z ) ) -> ( x e. RR <-> ( N ` ( T ` z ) ) e. RR ) )
8	6, 7	syl5ibr 236	. . . . . 6 |- ( x = ( N ` ( T ` z ) ) -> ( ( ( W e. NrmCVec /\ T : X --> Y ) /\ z e. X ) -> x e. RR ) )
9	8	impcom 446	. . . . 5 |- ( ( ( ( W e. NrmCVec /\ T : X --> Y ) /\ z e. X ) /\ x = ( N ` ( T ` z ) ) ) -> x e. RR )
10	9	adantrl 752	. . . 4 |- ( ( ( ( W e. NrmCVec /\ T : X --> Y ) /\ z e. X ) /\ ( ( M ` z ) <_ 1 /\ x = ( N ` ( T ` z ) ) ) ) -> x e. RR )
11	10	exp31 630	. . 3 |- ( ( W e. NrmCVec /\ T : X --> Y ) -> ( z e. X -> ( ( ( M ` z ) <_ 1 /\ x = ( N ` ( T ` z ) ) ) -> x e. RR ) ) )
12	11	rexlimdv 3030	. 2 |- ( ( W e. NrmCVec /\ T : X --> Y ) -> ( E. z e. X ( ( M ` z ) <_ 1 /\ x = ( N ` ( T ` z ) ) ) -> x e. RR ) )
13	12	abssdv 3676	1 |- ( ( W e. NrmCVec /\ T : X --> Y ) -> { x | E. z e. X ( ( M ` z ) <_ 1 /\ x = ( N ` ( T ` z ) ) ) } C_ RR )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   class class class wbr 4653   -->wf 5884   ` cfv 5888   RRcr 9935   1c1 9937   <_ cle 10075   NrmCVeccnv 27439   BaseSetcba 27441   norm_CVcnmcv 27445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  nmoxr  27621  nmooge0  27622  nmorepnf  27623  nmoolb  27626  nmoubi  27627  nmlno0lem  27648  nmopsetretHIL  28723