Theorem nmoeq0 22540

Description: The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypotheses

Ref	Expression
nmo0.1	|- N = ( S normOp T )
nmo0.2	|- V = ( Base ` S )
nmo0.3	|- .0. = ( 0g ` T )

Assertion

Ref	Expression
nmoeq0	|- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( ( N ` F ) = 0 <-> F = ( V X. { .0. } ) ) )

Proof of Theorem nmoeq0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . . 11 |- ( ( N ` F ) = 0 -> ( N ` F ) = 0 )
2		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
3	1, 2	syl6eqel 2709	. . . . . . . . . 10 |- ( ( N ` F ) = 0 -> ( N ` F ) e. RR )
4		nmo0.1	. . . . . . . . . . . 12 |- N = ( S normOp T )
5	4	isnghm2 22528	. . . . . . . . . . 11 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( F e. ( S NGHom T ) <-> ( N ` F ) e. RR ) )
6	5	biimpar 502	. . . . . . . . . 10 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) e. RR ) -> F e. ( S NGHom T ) )
7	3, 6	sylan2 491	. . . . . . . . 9 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> F e. ( S NGHom T ) )
8		nmo0.2	. . . . . . . . . 10 |- V = ( Base ` S )
9		eqid 2622	. . . . . . . . . 10 |- ( norm ` S ) = ( norm ` S )
10		eqid 2622	. . . . . . . . . 10 |- ( norm ` T ) = ( norm ` T )
11	4, 8, 9, 10	nmoi 22532	. . . . . . . . 9 |- ( ( F e. ( S NGHom T ) /\ x e. V ) -> ( ( norm ` T ) ` ( F ` x ) ) <_ ( ( N ` F ) x. ( ( norm ` S ) ` x ) ) )
12	7, 11	sylan 488	. . . . . . . 8 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` T ) ` ( F ` x ) ) <_ ( ( N ` F ) x. ( ( norm ` S ) ` x ) ) )
13		simplr 792	. . . . . . . . . 10 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( N ` F ) = 0 )
14	13	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( N ` F ) x. ( ( norm ` S ) ` x ) ) = ( 0 x. ( ( norm ` S ) ` x ) ) )
15		simpl1 1064	. . . . . . . . . . . 12 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> S e. NrmGrp )
16	8, 9	nmcl 22420	. . . . . . . . . . . 12 |- ( ( S e. NrmGrp /\ x e. V ) -> ( ( norm ` S ) ` x ) e. RR )
17	15, 16	sylan 488	. . . . . . . . . . 11 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` S ) ` x ) e. RR )
18	17	recnd 10068	. . . . . . . . . 10 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` S ) ` x ) e. CC )
19	18	mul02d 10234	. . . . . . . . 9 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( 0 x. ( ( norm ` S ) ` x ) ) = 0 )
20	14, 19	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( N ` F ) x. ( ( norm ` S ) ` x ) ) = 0 )
21	12, 20	breqtrd 4679	. . . . . . 7 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` T ) ` ( F ` x ) ) <_ 0 )
22		simpll2 1101	. . . . . . . 8 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> T e. NrmGrp )
23		simpl3 1066	. . . . . . . . . 10 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> F e. ( S GrpHom T ) )
24		eqid 2622	. . . . . . . . . . 11 |- ( Base ` T ) = ( Base ` T )
25	8, 24	ghmf 17664	. . . . . . . . . 10 |- ( F e. ( S GrpHom T ) -> F : V --> ( Base ` T ) )
26	23, 25	syl 17	. . . . . . . . 9 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> F : V --> ( Base ` T ) )
27	26	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( F ` x ) e. ( Base ` T ) )
28	24, 10	nmge0 22421	. . . . . . . 8 |- ( ( T e. NrmGrp /\ ( F ` x ) e. ( Base ` T ) ) -> 0 <_ ( ( norm ` T ) ` ( F ` x ) ) )
29	22, 27, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> 0 <_ ( ( norm ` T ) ` ( F ` x ) ) )
30	24, 10	nmcl 22420	. . . . . . . . 9 |- ( ( T e. NrmGrp /\ ( F ` x ) e. ( Base ` T ) ) -> ( ( norm ` T ) ` ( F ` x ) ) e. RR )
31	22, 27, 30	syl2anc 693	. . . . . . . 8 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` T ) ` ( F ` x ) ) e. RR )
32		letri3 10123	. . . . . . . 8 |- ( ( ( ( norm ` T ) ` ( F ` x ) ) e. RR /\ 0 e. RR ) -> ( ( ( norm ` T ) ` ( F ` x ) ) = 0 <-> ( ( ( norm ` T ) ` ( F ` x ) ) <_ 0 /\ 0 <_ ( ( norm ` T ) ` ( F ` x ) ) ) ) )
33	31, 2, 32	sylancl 694	. . . . . . 7 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( ( norm ` T ) ` ( F ` x ) ) = 0 <-> ( ( ( norm ` T ) ` ( F ` x ) ) <_ 0 /\ 0 <_ ( ( norm ` T ) ` ( F ` x ) ) ) ) )
34	21, 29, 33	mpbir2and 957	. . . . . 6 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( norm ` T ) ` ( F ` x ) ) = 0 )
35		nmo0.3	. . . . . . . 8 |- .0. = ( 0g ` T )
36	24, 10, 35	nmeq0 22422	. . . . . . 7 |- ( ( T e. NrmGrp /\ ( F ` x ) e. ( Base ` T ) ) -> ( ( ( norm ` T ) ` ( F ` x ) ) = 0 <-> ( F ` x ) = .0. ) )
37	22, 27, 36	syl2anc 693	. . . . . 6 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( ( ( norm ` T ) ` ( F ` x ) ) = 0 <-> ( F ` x ) = .0. ) )
38	34, 37	mpbid 222	. . . . 5 |- ( ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) /\ x e. V ) -> ( F ` x ) = .0. )
39	38	mpteq2dva 4744	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> ( x e. V |-> ( F ` x ) ) = ( x e. V |-> .0. ) )
40	26	feqmptd 6249	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> F = ( x e. V |-> ( F ` x ) ) )
41		fconstmpt 5163	. . . . 5 |- ( V X. { .0. } ) = ( x e. V |-> .0. )
42	41	a1i 11	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> ( V X. { .0. } ) = ( x e. V |-> .0. ) )
43	39, 40, 42	3eqtr4d 2666	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( N ` F ) = 0 ) -> F = ( V X. { .0. } ) )
44	43	ex 450	. 2 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( ( N ` F ) = 0 -> F = ( V X. { .0. } ) ) )
45	4, 8, 35	nmo0 22539	. . . 4 |- ( ( S e. NrmGrp /\ T e. NrmGrp ) -> ( N ` ( V X. { .0. } ) ) = 0 )
46	45	3adant3 1081	. . 3 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( N ` ( V X. { .0. } ) ) = 0 )
47		fveq2 6191	. . . 4 |- ( F = ( V X. { .0. } ) -> ( N ` F ) = ( N ` ( V X. { .0. } ) ) )
48	47	eqeq1d 2624	. . 3 |- ( F = ( V X. { .0. } ) -> ( ( N ` F ) = 0 <-> ( N ` ( V X. { .0. } ) ) = 0 ) )
49	46, 48	syl5ibrcom 237	. 2 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( F = ( V X. { .0. } ) -> ( N ` F ) = 0 ) )
50	44, 49	impbid 202	1 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( ( N ` F ) = 0 <-> F = ( V X. { .0. } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   x. cmul 9941   <_ cle 10075   Basecbs 15857   0gc0g 16100   GrpHom cghm 17657   normcnm 22381  NrmGrpcngp 22382   normOpcnmo 22509   NGHom cnghm 22510

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-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nmo 22512  df-nghm 22513

This theorem is referenced by: (None)