Theorem nmlnop0iALT 23503

Description: A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nmlnop0.1	|- T e. LinOp

Assertion

Ref	Expression
nmlnop0iALT	|- ( ( normop ` T ) = 0 <-> T = 0hop )

Proof of Theorem nmlnop0iALT

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

Step	Hyp	Ref	Expression
1		normcl 22632	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( normh ` x ) e. RR )
2	1	recnd 9119	. . . . . . . . . . . . 13 |- ( x e. ~H -> ( normh ` x ) e. CC )
3	2	adantr 453	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` x ) e. CC )
4		norm-i 22636	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( normh ` x ) = 0 <-> x = 0h ) )
5		fveq2 5731	. . . . . . . . . . . . . . . 16 |- ( x = 0h -> ( T ` x ) = ( T ` 0h ) )
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 |- T e. LinOp
7	6	lnop0i 23478	. . . . . . . . . . . . . . . 16 |- ( T ` 0h ) = 0h
8	5, 7	syl6eq 2486	. . . . . . . . . . . . . . 15 |- ( x = 0h -> ( T ` x ) = 0h )
9	4, 8	syl6bi 221	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( ( normh ` x ) = 0 -> ( T ` x ) = 0h ) )
10	9	necon3d 2641	. . . . . . . . . . . . 13 |- ( x e. ~H -> ( ( T ` x ) =/= 0h -> ( normh ` x ) =/= 0 ) )
11	10	imp 420	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` x ) =/= 0 )
12	3, 11	recne0d 9789	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( 1 / ( normh ` x ) ) =/= 0 )
13		simpr 449	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` x ) =/= 0h )
14	3, 11	reccld 9788	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( 1 / ( normh ` x ) ) e. CC )
15	6	lnopfi 23477	. . . . . . . . . . . . . . . 16 |- T : ~H --> ~H
16	15	ffvelrni 5872	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( T ` x ) e. ~H )
17	16	adantr 453	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` x ) e. ~H )
18		hvmul0or 22532	. . . . . . . . . . . . . 14 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ ( T ` x ) e. ~H ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) = 0h <-> ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) ) )
19	14, 17, 18	syl2anc 644	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) = 0h <-> ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) ) )
20	19	necon3abid 2636	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h <-> -. ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) ) )
21		neanior 2691	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) =/= 0 /\ ( T ` x ) =/= 0h ) <-> -. ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) )
22	20, 21	syl6bbr 256	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h <-> ( ( 1 / ( normh ` x ) ) =/= 0 /\ ( T ` x ) =/= 0h ) ) )
23	12, 13, 22	mpbir2and 890	. . . . . . . . . 10 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h )
24		hvmulcl 22521	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ ( T ` x ) e. ~H ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H )
25	14, 17, 24	syl2anc 644	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H )
26		normgt0 22634	. . . . . . . . . . 11 |- ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h <-> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
27	25, 26	syl 16	. . . . . . . . . 10 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h <-> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
28	23, 27	mpbid 203	. . . . . . . . 9 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) )
29	28	ex 425	. . . . . . . 8 |- ( x e. ~H -> ( ( T ` x ) =/= 0h -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
30	29	adantl 454	. . . . . . 7 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( ( T ` x ) =/= 0h -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
31		nmopsetretHIL 23372	. . . . . . . . . . . . . 14 |- ( T : ~H --> ~H -> { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR )
32	15, 31	ax-mp 5	. . . . . . . . . . . . 13 |- { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR
33		ressxr 9134	. . . . . . . . . . . . 13 |- RR C_ RR*
34	32, 33	sstri 3359	. . . . . . . . . . . 12 |- { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR*
35		simpl 445	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> x e. ~H )
36		hvmulcl 22521	. . . . . . . . . . . . . . 15 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ x e. ~H ) -> ( ( 1 / ( normh ` x ) ) .h x ) e. ~H )
37	14, 35, 36	syl2anc 644	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h x ) e. ~H )
38	8	necon3i 2645	. . . . . . . . . . . . . . . . 17 |- ( ( T ` x ) =/= 0h -> x =/= 0h )
39		norm1 22756	. . . . . . . . . . . . . . . . 17 |- ( ( x e. ~H /\ x =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) = 1 )
40	38, 39	sylan2 462	. . . . . . . . . . . . . . . 16 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) = 1 )
41		1re 9095	. . . . . . . . . . . . . . . 16 |- 1 e. RR
42	40, 41	syl6eqel 2526	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) e. RR )
43		eqle 9181	. . . . . . . . . . . . . . 15 |- ( ( ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) e. RR /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) = 1 ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 )
44	42, 40, 43	syl2anc 644	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 )
45	6	lnopmuli 23480	. . . . . . . . . . . . . . . . 17 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ x e. ~H ) -> ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) = ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) )
46	14, 35, 45	syl2anc 644	. . . . . . . . . . . . . . . 16 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) = ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) )
47	46	eqcomd 2443	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) = ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
48	47	fveq2d 5735	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) )
49		fveq2 5731	. . . . . . . . . . . . . . . . 17 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( normh ` z ) = ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
50	49	breq1d 4225	. . . . . . . . . . . . . . . 16 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( ( normh ` z ) <_ 1 <-> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 ) )
51		fveq2 5731	. . . . . . . . . . . . . . . . . 18 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( T ` z ) = ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
52	51	fveq2d 5735	. . . . . . . . . . . . . . . . 17 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( normh ` ( T ` z ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) )
53	52	eqeq2d 2449	. . . . . . . . . . . . . . . 16 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) <-> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) ) )
54	50, 53	anbi12d 693	. . . . . . . . . . . . . . 15 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) <-> ( ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) ) ) )
55	54	rspcev 3054	. . . . . . . . . . . . . 14 |- ( ( ( ( 1 / ( normh ` x ) ) .h x ) e. ~H /\ ( ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) ) ) -> E. z e. ~H ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) )
56	37, 44, 48, 55	syl12anc 1183	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> E. z e. ~H ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) )
57		fvex 5745	. . . . . . . . . . . . . 14 |- ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. _V
58		eqeq1 2444	. . . . . . . . . . . . . . . 16 |- ( y = ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) -> ( y = ( normh ` ( T ` z ) ) <-> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) )
59	58	anbi2d 686	. . . . . . . . . . . . . . 15 |- ( y = ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) -> ( ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) <-> ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) ) )
60	59	rexbidv 2728	. . . . . . . . . . . . . 14 |- ( y = ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) -> ( E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) <-> E. z e. ~H ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) ) )
61	57, 60	elab 3084	. . . . . . . . . . . . 13 |- ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } <-> E. z e. ~H ( ( normh ` z ) <_ 1 /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` z ) ) ) )
62	56, 61	sylibr 205	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } )
63		supxrub 10908	. . . . . . . . . . . 12 |- ( ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR* /\ ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) )
64	34, 62, 63	sylancr 646	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) )
65	64	adantll 696	. . . . . . . . . 10 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) )
66		nmopval 23364	. . . . . . . . . . . . . 14 |- ( T : ~H --> ~H -> ( normop ` T ) = sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) )
67	15, 66	ax-mp 5	. . . . . . . . . . . . 13 |- ( normop ` T ) = sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < )
68	67	eqeq1i 2445	. . . . . . . . . . . 12 |- ( ( normop ` T ) = 0 <-> sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) = 0 )
69	68	biimpi 188	. . . . . . . . . . 11 |- ( ( normop ` T ) = 0 -> sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) = 0 )
70	69	ad2antrr 708	. . . . . . . . . 10 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) = 0 )
71	65, 70	breqtrd 4239	. . . . . . . . 9 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ 0 )
72		normcl 22632	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. RR )
73	25, 72	syl 16	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. RR )
74		0re 9096	. . . . . . . . . . 11 |- 0 e. RR
75		lenlt 9159	. . . . . . . . . . 11 |- ( ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. RR /\ 0 e. RR ) -> ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ 0 <-> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
76	73, 74, 75	sylancl 645	. . . . . . . . . 10 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ 0 <-> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
77	76	adantll 696	. . . . . . . . 9 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> ( ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ 0 <-> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
78	71, 77	mpbid 203	. . . . . . . 8 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) )
79	78	ex 425	. . . . . . 7 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( ( T ` x ) =/= 0h -> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
80	30, 79	pm2.65d 169	. . . . . 6 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> -. ( T ` x ) =/= 0h )
81		nne 2607	. . . . . 6 |- ( -. ( T ` x ) =/= 0h <-> ( T ` x ) = 0h )
82	80, 81	sylib 190	. . . . 5 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( T ` x ) = 0h )
83		ho0val 23258	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
84	83	adantl 454	. . . . 5 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( 0hop ` x ) = 0h )
85	82, 84	eqtr4d 2473	. . . 4 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( T ` x ) = ( 0hop ` x ) )
86	85	ralrimiva 2791	. . 3 |- ( ( normop ` T ) = 0 -> A. x e. ~H ( T ` x ) = ( 0hop ` x ) )
87		ffn 5594	. . . . 5 |- ( T : ~H --> ~H -> T Fn ~H )
88	15, 87	ax-mp 5	. . . 4 |- T Fn ~H
89		ho0f 23259	. . . . 5 |- 0hop : ~H --> ~H
90		ffn 5594	. . . . 5 |- ( 0hop : ~H --> ~H -> 0hop Fn ~H )
91	89, 90	ax-mp 5	. . . 4 |- 0hop Fn ~H
92		eqfnfv 5830	. . . 4 |- ( ( T Fn ~H /\ 0hop Fn ~H ) -> ( T = 0hop <-> A. x e. ~H ( T ` x ) = ( 0hop ` x ) ) )
93	88, 91, 92	mp2an 655	. . 3 |- ( T = 0hop <-> A. x e. ~H ( T ` x ) = ( 0hop ` x ) )
94	86, 93	sylibr 205	. 2 |- ( ( normop ` T ) = 0 -> T = 0hop )
95		fveq2 5731	. . 3 |- ( T = 0hop -> ( normop ` T ) = ( normop ` 0hop ) )
96		nmop0 23494	. . 3 |- ( normop ` 0hop ) = 0
97	95, 96	syl6eq 2486	. 2 |- ( T = 0hop -> ( normop ` T ) = 0 )
98	94, 97	impbii 182	1 |- ( ( normop ` T ) = 0 <-> T = 0hop )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   \/ wo 359   /\ wa 360   = wceq 1653   e. wcel 1726   {cab 2424   =/= wne 2601   A.wral 2707   E.wrex 2708   C_ wss 3322   class class class wbr 4215   Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   RR*cxr 9124   < clt 9125   <_ cle 9126   / cdiv 9682   ~Hchil 22427   .h csm 22429   normhcno 22431   0hc0v 22432   0hopch0o 22451   normopcnop 22453   LinOpclo 22455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cc 8320  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592  ax-hcompl 22709

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-cn 17296  df-cnp 17297  df-lm 17298  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cfil 19213  df-cau 19214  df-cmet 19215  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-subgo 21895  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-dip 22202  df-ssp 22226  df-ph 22319  df-cbn 22370  df-hnorm 22476  df-hba 22477  df-hvsub 22479  df-hlim 22480  df-hcau 22481  df-sh 22714  df-ch 22729  df-oc 22759  df-ch0 22760  df-shs 22815  df-pjh 22902  df-h0op 23256  df-nmop 23347  df-lnop 23349