Theorem nmoo0 27646

Description: The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoo0.3	|- N = ( U normOpOLD W )
nmoo0.0	|- Z = ( U 0op W )

Assertion

Ref	Expression
nmoo0	|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` Z ) = 0 )

Proof of Theorem nmoo0

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( BaseSet ` U ) = ( BaseSet ` U )
2		eqid 2622	. . . . 5 |- ( BaseSet ` W ) = ( BaseSet ` W )
3		nmoo0.0	. . . . 5 |- Z = ( U 0op W )
4	1, 2, 3	0oo 27644	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> Z : ( BaseSet ` U ) --> ( BaseSet ` W ) )
5		eqid 2622	. . . . 5 |- ( normCV ` U ) = ( normCV ` U )
6		eqid 2622	. . . . 5 |- ( normCV ` W ) = ( normCV ` W )
7		nmoo0.3	. . . . 5 |- N = ( U normOpOLD W )
8	1, 2, 5, 6, 7	nmooval 27618	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ Z : ( BaseSet ` U ) --> ( BaseSet ` W ) ) -> ( N ` Z ) = sup ( { x | E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) } , RR* , < ) )
9	4, 8	mpd3an3 1425	. . 3 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` Z ) = sup ( { x | E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) } , RR* , < ) )
10		df-sn 4178	. . . . 5 |- { 0 } = { x | x = 0 }
11		eqid 2622	. . . . . . . . . . 11 |- ( 0vec ` U ) = ( 0vec ` U )
12	1, 11	nvzcl 27489	. . . . . . . . . 10 |- ( U e. NrmCVec -> ( 0vec ` U ) e. ( BaseSet ` U ) )
13	11, 5	nvz0 27523	. . . . . . . . . . 11 |- ( U e. NrmCVec -> ( ( normCV ` U ) ` ( 0vec ` U ) ) = 0 )
14		0le1 10551	. . . . . . . . . . 11 |- 0 <_ 1
15	13, 14	syl6eqbr 4692	. . . . . . . . . 10 |- ( U e. NrmCVec -> ( ( normCV ` U ) ` ( 0vec ` U ) ) <_ 1 )
16		fveq2 6191	. . . . . . . . . . . 12 |- ( z = ( 0vec ` U ) -> ( ( normCV ` U ) ` z ) = ( ( normCV ` U ) ` ( 0vec ` U ) ) )
17	16	breq1d 4663	. . . . . . . . . . 11 |- ( z = ( 0vec ` U ) -> ( ( ( normCV ` U ) ` z ) <_ 1 <-> ( ( normCV ` U ) ` ( 0vec ` U ) ) <_ 1 ) )
18	17	rspcev 3309	. . . . . . . . . 10 |- ( ( ( 0vec ` U ) e. ( BaseSet ` U ) /\ ( ( normCV ` U ) ` ( 0vec ` U ) ) <_ 1 ) -> E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 )
19	12, 15, 18	syl2anc 693	. . . . . . . . 9 |- ( U e. NrmCVec -> E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 )
20	19	biantrurd 529	. . . . . . . 8 |- ( U e. NrmCVec -> ( x = 0 <-> ( E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) ) )
21	20	adantr 481	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( x = 0 <-> ( E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) ) )
22		eqid 2622	. . . . . . . . . . . . . . 15 |- ( 0vec ` W ) = ( 0vec ` W )
23	1, 22, 3	0oval 27643	. . . . . . . . . . . . . 14 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ z e. ( BaseSet ` U ) ) -> ( Z ` z ) = ( 0vec ` W ) )
24	23	3expa 1265	. . . . . . . . . . . . 13 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( Z ` z ) = ( 0vec ` W ) )
25	24	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( ( normCV ` W ) ` ( Z ` z ) ) = ( ( normCV ` W ) ` ( 0vec ` W ) ) )
26	22, 6	nvz0 27523	. . . . . . . . . . . . 13 |- ( W e. NrmCVec -> ( ( normCV ` W ) ` ( 0vec ` W ) ) = 0 )
27	26	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( ( normCV ` W ) ` ( 0vec ` W ) ) = 0 )
28	25, 27	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( ( normCV ` W ) ` ( Z ` z ) ) = 0 )
29	28	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( x = ( ( normCV ` W ) ` ( Z ` z ) ) <-> x = 0 ) )
30	29	anbi2d 740	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ z e. ( BaseSet ` U ) ) -> ( ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) <-> ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) ) )
31	30	rexbidva 3049	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) <-> E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) ) )
32		r19.41v 3089	. . . . . . . 8 |- ( E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) <-> ( E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) )
33	31, 32	syl6rbb 277	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( ( E. z e. ( BaseSet ` U ) ( ( normCV ` U ) ` z ) <_ 1 /\ x = 0 ) <-> E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) ) )
34	21, 33	bitrd 268	. . . . . 6 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( x = 0 <-> E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) ) )
35	34	abbidv 2741	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> { x | x = 0 } = { x | E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) } )
36	10, 35	syl5req 2669	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> { x | E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) } = { 0 } )
37	36	supeq1d 8352	. . 3 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> sup ( { x | E. z e. ( BaseSet ` U ) ( ( ( normCV ` U ) ` z ) <_ 1 /\ x = ( ( normCV ` W ) ` ( Z ` z ) ) ) } , RR* , < ) = sup ( { 0 } , RR* , < ) )
38	9, 37	eqtrd 2656	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` Z ) = sup ( { 0 } , RR* , < ) )
39		xrltso 11974	. . 3 |- < Or RR*
40		0xr 10086	. . 3 |- 0 e. RR*
41		supsn 8378	. . 3 |- ( ( < Or RR* /\ 0 e. RR* ) -> sup ( { 0 } , RR* , < ) = 0 )
42	39, 40, 41	mp2an 708	. 2 |- sup ( { 0 } , RR* , < ) = 0
43	38, 42	syl6eq 2672	1 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` Z ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {csn 4177   class class class wbr 4653   Or wor 5034   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NrmCVeccnv 27439   BaseSetcba 27441   0veccn0v 27443   norm_CVcnmcv 27445   normOpOLDcnmoo 27596   0op c0o 27598

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-nmoo 27600  df-0o 27602

This theorem is referenced by:  0blo  27647  nmlno0lem  27648