Theorem nmlnop0iALT 22575

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 21704	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( normh ` x ) e. RR )
2	1	recnd 8861	. . . . . . . . . . . . 13 |- ( x e. ~H -> ( normh ` x ) e. CC )
3	2	adantr 451	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` x ) e. CC )
4		norm-i 21708	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( normh ` x ) = 0 <-> x = 0h ) )
5		fveq2 5525	. . . . . . . . . . . . . . . 16 |- ( x = 0h -> ( T ` x ) = ( T ` 0h ) )
6		nmlnop0.1	. . . . . . . . . . . . . . . . 17 |- T e. LinOp
7	6	lnop0i 22550	. . . . . . . . . . . . . . . 16 |- ( T ` 0h ) = 0h
8	5, 7	syl6eq 2331	. . . . . . . . . . . . . . 15 |- ( x = 0h -> ( T ` x ) = 0h )
9	4, 8	syl6bi 219	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( ( normh ` x ) = 0 -> ( T ` x ) = 0h ) )
10	9	necon3d 2484	. . . . . . . . . . . . 13 |- ( x e. ~H -> ( ( T ` x ) =/= 0h -> ( normh ` x ) =/= 0 ) )
11	10	imp 418	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` x ) =/= 0 )
12	3, 11	recne0d 9530	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( 1 / ( normh ` x ) ) =/= 0 )
13		simpr 447	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` x ) =/= 0h )
14	3, 11	reccld 9529	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( 1 / ( normh ` x ) ) e. CC )
15	6	lnopfi 22549	. . . . . . . . . . . . . . . 16 |- T : ~H --> ~H
16	15	ffvelrni 5664	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( T ` x ) e. ~H )
17	16	adantr 451	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` x ) e. ~H )
18		hvmul0or 21604	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) = 0h <-> ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) ) )
20	19	necon3abid 2479	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h <-> -. ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) ) )
21		neanior 2531	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) =/= 0 /\ ( T ` x ) =/= 0h ) <-> -. ( ( 1 / ( normh ` x ) ) = 0 \/ ( T ` x ) = 0h ) )
22	20, 21	syl6bbr 254	. . . . . . . . . . 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 888	. . . . . . . . . 10 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) =/= 0h )
24		hvmulcl 21593	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ ( T ` x ) e. ~H ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H )
25	14, 17, 24	syl2anc 642	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H )
26		normgt0 21706	. . . . . . . . . . 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 15	. . . . . . . . . 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 201	. . . . . . . . 9 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) )
29	28	ex 423	. . . . . . . 8 |- ( x e. ~H -> ( ( T ` x ) =/= 0h -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
30	29	adantl 452	. . . . . . 7 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( ( T ` x ) =/= 0h -> 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
31		nmopsetretHIL 22444	. . . . . . . . . . . . . 14 |- ( T : ~H --> ~H -> { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR )
32	15, 31	ax-mp 8	. . . . . . . . . . . . 13 |- { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR
33		ressxr 8876	. . . . . . . . . . . . 13 |- RR C_ RR*
34	32, 33	sstri 3188	. . . . . . . . . . . 12 |- { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } C_ RR*
35		simpl 443	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> x e. ~H )
36		hvmulcl 21593	. . . . . . . . . . . . . . 15 |- ( ( ( 1 / ( normh ` x ) ) e. CC /\ x e. ~H ) -> ( ( 1 / ( normh ` x ) ) .h x ) e. ~H )
37	14, 35, 36	syl2anc 642	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h x ) e. ~H )
38	8	necon3i 2485	. . . . . . . . . . . . . . . . 17 |- ( ( T ` x ) =/= 0h -> x =/= 0h )
39		norm1 21828	. . . . . . . . . . . . . . . . 17 |- ( ( x e. ~H /\ x =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) = 1 )
40	38, 39	sylan2 460	. . . . . . . . . . . . . . . 16 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) = 1 )
41		1re 8837	. . . . . . . . . . . . . . . 16 |- 1 e. RR
42	40, 41	syl6eqel 2371	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) e. RR )
43		eqle 8923	. . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 )
45	6	lnopmuli 22552	. . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . 16 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) = ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) )
47	46	eqcomd 2288	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) = ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
48	47	fveq2d 5529	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) )
49		fveq2 5525	. . . . . . . . . . . . . . . . 17 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( normh ` z ) = ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
50	49	breq1d 4033	. . . . . . . . . . . . . . . 16 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( ( normh ` z ) <_ 1 <-> ( normh ` ( ( 1 / ( normh ` x ) ) .h x ) ) <_ 1 ) )
51		fveq2 5525	. . . . . . . . . . . . . . . . . 18 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( T ` z ) = ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) )
52	51	fveq2d 5529	. . . . . . . . . . . . . . . . 17 |- ( z = ( ( 1 / ( normh ` x ) ) .h x ) -> ( normh ` ( T ` z ) ) = ( normh ` ( T ` ( ( 1 / ( normh ` x ) ) .h x ) ) ) )
53	52	eqeq2d 2294	. . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . 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 2884	. . . . . . . . . . . . . 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 1180	. . . . . . . . . . . . 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 5539	. . . . . . . . . . . . . 14 |- ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. _V
58		eqeq1 2289	. . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . 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 2564	. . . . . . . . . . . . . 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 2914	. . . . . . . . . . . . 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 203	. . . . . . . . . . . 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 10643	. . . . . . . . . . . 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 644	. . . . . . . . . . 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 694	. . . . . . . . . 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 22436	. . . . . . . . . . . . . 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 8	. . . . . . . . . . . . 13 |- ( normop ` T ) = sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < )
68	67	eqeq1i 2290	. . . . . . . . . . . 12 |- ( ( normop ` T ) = 0 <-> sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) = 0 )
69	68	biimpi 186	. . . . . . . . . . 11 |- ( ( normop ` T ) = 0 -> sup ( { y | E. z e. ~H ( ( normh ` z ) <_ 1 /\ y = ( normh ` ( T ` z ) ) ) } , RR* , < ) = 0 )
70	69	ad2antrr 706	. . . . . . . . . 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 4047	. . . . . . . . 9 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) <_ 0 )
72		normcl 21704	. . . . . . . . . . . 12 |- ( ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) e. ~H -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. RR )
73	25, 72	syl 15	. . . . . . . . . . 11 |- ( ( x e. ~H /\ ( T ` x ) =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) e. RR )
74		0re 8838	. . . . . . . . . . 11 |- 0 e. RR
75		lenlt 8901	. . . . . . . . . . 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 643	. . . . . . . . . 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 694	. . . . . . . . 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 201	. . . . . . . 8 |- ( ( ( ( normop ` T ) = 0 /\ x e. ~H ) /\ ( T ` x ) =/= 0h ) -> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) )
79	78	ex 423	. . . . . . 7 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( ( T ` x ) =/= 0h -> -. 0 < ( normh ` ( ( 1 / ( normh ` x ) ) .h ( T ` x ) ) ) ) )
80	30, 79	pm2.65d 166	. . . . . 6 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> -. ( T ` x ) =/= 0h )
81		nne 2450	. . . . . 6 |- ( -. ( T ` x ) =/= 0h <-> ( T ` x ) = 0h )
82	80, 81	sylib 188	. . . . 5 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( T ` x ) = 0h )
83		ho0val 22330	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
84	83	adantl 452	. . . . 5 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( 0hop ` x ) = 0h )
85	82, 84	eqtr4d 2318	. . . 4 |- ( ( ( normop ` T ) = 0 /\ x e. ~H ) -> ( T ` x ) = ( 0hop ` x ) )
86	85	ralrimiva 2626	. . 3 |- ( ( normop ` T ) = 0 -> A. x e. ~H ( T ` x ) = ( 0hop ` x ) )
87		ffn 5389	. . . . 5 |- ( T : ~H --> ~H -> T Fn ~H )
88	15, 87	ax-mp 8	. . . 4 |- T Fn ~H
89		ho0f 22331	. . . . 5 |- 0hop : ~H --> ~H
90		ffn 5389	. . . . 5 |- ( 0hop : ~H --> ~H -> 0hop Fn ~H )
91	89, 90	ax-mp 8	. . . 4 |- 0hop Fn ~H
92		eqfnfv 5622	. . . 4 |- ( ( T Fn ~H /\ 0hop Fn ~H ) -> ( T = 0hop <-> A. x e. ~H ( T ` x ) = ( 0hop ` x ) ) )
93	88, 91, 92	mp2an 653	. . 3 |- ( T = 0hop <-> A. x e. ~H ( T ` x ) = ( 0hop ` x ) )
94	86, 93	sylibr 203	. 2 |- ( ( normop ` T ) = 0 -> T = 0hop )
95		fveq2 5525	. . 3 |- ( T = 0hop -> ( normop ` T ) = ( normop ` 0hop ) )
96		nmop0 22566	. . 3 |- ( normop ` 0hop ) = 0
97	95, 96	syl6eq 2331	. 2 |- ( T = 0hop -> ( normop ` T ) = 0 )
98	94, 97	impbii 180	1 |- ( ( normop ` T ) = 0 <-> T = 0hop )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1623   e. wcel 1684   {cab 2269   =/= wne 2446   A.wral 2543   E.wrex 2544   C_ wss 3152   class class class wbr 4023   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866   < clt 8867   <_ cle 8868   / cdiv 9423   ~Hchil 21499   .h csm 21501   normhcno 21503   0hc0v 21504   0hopch0o 21523   normopcnop 21525   LinOpclo 21527

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-pjh 21974  df-h0op 22328  df-nmop 22419  df-lnop 22421