Theorem hmopidmchi 25474

Description: An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hmopidmch.1	|- T e. HrmOp
hmopidmch.2	|- ( T o. T ) = T

Assertion

Ref	Expression
hmopidmchi	|- ran T e. CH

Proof of Theorem hmopidmchi

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

Step	Hyp	Ref	Expression
1		hmopidmch.1	. . . 4 |- T e. HrmOp
2		hmoplin 25265	. . . 4 |- ( T e. HrmOp -> T e. LinOp )
3	1, 2	ax-mp 5	. . 3 |- T e. LinOp
4	3	rnelshi 25382	. 2 |- ran T e. SH
5		eqid 2441	. . . . . . . 8 |- ( normh o. -h ) = ( normh o. -h )
6	5	hilxmet 24516	. . . . . . 7 |- ( normh o. -h ) e. ( *Met ` ~H )
7		eqid 2441	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
8	7	methaus 19995	. . . . . . 7 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. Haus )
9	6, 8	mp1i 12	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( MetOpen ` ( normh o. -h ) ) e. Haus )
10		eqid 2441	. . . . . . . . . . . 12 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
11	10, 5	hhims 24493	. . . . . . . . . . . 12 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12	10, 11, 7	hhlm 24520	. . . . . . . . . . 11 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
13		resss 5131	. . . . . . . . . . 11 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
14	12, 13	eqsstri 3383	. . . . . . . . . 10 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	14	ssbri 4331	. . . . . . . . 9 |- ( f ~~>v x -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
16	15	adantl 463	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
17	7	mopntopon 19914	. . . . . . . . . 10 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
18	6, 17	mp1i 12	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
19	3	lnopfi 25292	. . . . . . . . . . . 12 |- T : ~H --> ~H
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T : ~H --> ~H )
21	20	feqmptd 5741	. . . . . . . . . 10 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T = ( y e. ~H |-> ( T ` y ) ) )
22		hmopbdoptHIL 25311	. . . . . . . . . . . . 13 |- ( T e. HrmOp -> T e. BndLinOp )
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 |- T e. BndLinOp
24		lnopcnbd 25359	. . . . . . . . . . . . 13 |- ( T e. LinOp -> ( T e. ConOp <-> T e. BndLinOp ) )
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 |- ( T e. ConOp <-> T e. BndLinOp )
26	23, 25	mpbir 209	. . . . . . . . . . 11 |- T e. ConOp
27	5, 7	hhcno 25227	. . . . . . . . . . 11 |- ConOp = ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
28	26, 27	eleqtri 2513	. . . . . . . . . 10 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
29	21, 28	syl6eqelr 2530	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( y e. ~H |-> ( T ` y ) ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
30	18	cnmptid 19134	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( y e. ~H |-> y ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
31	10	hhnv 24486	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. e. NrmCVec
32	10	hhvs 24491	. . . . . . . . . . 11 |- -h = ( -v ` <. <. +h , .h >. , normh >. )
33	11, 7, 32	vmcn 24013	. . . . . . . . . 10 |- ( <. <. +h , .h >. , normh >. e. NrmCVec -> -h e. ( ( ( MetOpen ` ( normh o. -h ) ) tX ( MetOpen ` ( normh o. -h ) ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
34	31, 33	mp1i 12	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> -h e. ( ( ( MetOpen ` ( normh o. -h ) ) tX ( MetOpen ` ( normh o. -h ) ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
35	18, 29, 30, 34	cnmpt12f 19139	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( y e. ~H |-> ( ( T ` y ) -h y ) ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
36	16, 35	lmcn 18809	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` x ) )
37		simpl 454	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ran T )
38	4	shssii 24534	. . . . . . . . . . . . . 14 |- ran T C_ ~H
39		fss 5564	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ ran T C_ ~H ) -> f : NN --> ~H )
40	37, 38, 39	sylancl 657	. . . . . . . . . . . . 13 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ~H )
41	40	ffvelrnda 5840	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ~H )
42		fveq2 5688	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( T ` y ) = ( T ` ( f ` k ) ) )
43		id 22	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> y = ( f ` k ) )
44	42, 43	oveq12d 6108	. . . . . . . . . . . . 13 |- ( y = ( f ` k ) -> ( ( T ` y ) -h y ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
45		eqid 2441	. . . . . . . . . . . . 13 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) = ( y e. ~H |-> ( ( T ` y ) -h y ) )
46		ovex 6115	. . . . . . . . . . . . 13 |- ( ( T ` ( f ` k ) ) -h ( f ` k ) ) e. _V
47	44, 45, 46	fvmpt 5771	. . . . . . . . . . . 12 |- ( ( f ` k ) e. ~H -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
48	41, 47	syl 16	. . . . . . . . . . 11 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
49		ffn 5556	. . . . . . . . . . . . . . . 16 |- ( T : ~H --> ~H -> T Fn ~H )
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- T Fn ~H
51		fveq2 5688	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> ( T ` y ) = ( T ` ( T ` x ) ) )
52		id 22	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> y = ( T ` x ) )
53	51, 52	eqeq12d 2455	. . . . . . . . . . . . . . . 16 |- ( y = ( T ` x ) -> ( ( T ` y ) = y <-> ( T ` ( T ` x ) ) = ( T ` x ) ) )
54	53	ralrn 5843	. . . . . . . . . . . . . . 15 |- ( T Fn ~H -> ( A. y e. ran T ( T ` y ) = y <-> A. x e. ~H ( T ` ( T ` x ) ) = ( T ` x ) ) )
55	50, 54	ax-mp 5	. . . . . . . . . . . . . 14 |- ( A. y e. ran T ( T ` y ) = y <-> A. x e. ~H ( T ` ( T ` x ) ) = ( T ` x ) )
56		hmopidmch.2	. . . . . . . . . . . . . . . 16 |- ( T o. T ) = T
57	56	fveq1i 5689	. . . . . . . . . . . . . . 15 |- ( ( T o. T ) ` x ) = ( T ` x )
58	19, 19	hocoi 25087	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( T o. T ) ` x ) = ( T ` ( T ` x ) ) )
59	57, 58	syl5reqr 2488	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( T ` ( T ` x ) ) = ( T ` x ) )
60	55, 59	mprgbir 2784	. . . . . . . . . . . . 13 |- A. y e. ran T ( T ` y ) = y
61		ffvelrn 5838	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ k e. NN ) -> ( f ` k ) e. ran T )
62	61	adantlr 709	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ran T )
63	42, 43	eqeq12d 2455	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( ( T ` y ) = y <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
64	63	rspccv 3067	. . . . . . . . . . . . 13 |- ( A. y e. ran T ( T ` y ) = y -> ( ( f ` k ) e. ran T -> ( T ` ( f ` k ) ) = ( f ` k ) ) )
65	60, 62, 64	mpsyl 63	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) = ( f ` k ) )
66	65, 41	eqeltrd 2515	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) e. ~H )
67		hvsubeq0 24389	. . . . . . . . . . . . 13 |- ( ( ( T ` ( f ` k ) ) e. ~H /\ ( f ` k ) e. ~H ) -> ( ( ( T ` ( f ` k ) ) -h ( f ` k ) ) = 0h <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
68	66, 41, 67	syl2anc 656	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( ( T ` ( f ` k ) ) -h ( f ` k ) ) = 0h <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
69	65, 68	mpbird 232	. . . . . . . . . . 11 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( T ` ( f ` k ) ) -h ( f ` k ) ) = 0h )
70	48, 69	eqtrd 2473	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = 0h )
71		fvco3 5765	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ k e. NN ) -> ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) )
72	71	adantlr 709	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) )
73		ax-hv0cl 24324	. . . . . . . . . . . . 13 |- 0h e. ~H
74	73	elexi 2980	. . . . . . . . . . . 12 |- 0h e. _V
75	74	fvconst2 5930	. . . . . . . . . . 11 |- ( k e. NN -> ( ( NN X. { 0h } ) ` k ) = 0h )
76	75	adantl 463	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( NN X. { 0h } ) ` k ) = 0h )
77	70, 72, 76	3eqtr4d 2483	. . . . . . . . 9 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( NN X. { 0h } ) ` k ) )
78	77	ralrimiva 2797	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> A. k e. NN ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( NN X. { 0h } ) ` k ) )
79		ovex 6115	. . . . . . . . . . 11 |- ( ( T ` y ) -h y ) e. _V
80	79, 45	fnmpti 5536	. . . . . . . . . 10 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H
81		fnfco 5574	. . . . . . . . . 10 |- ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H /\ f : NN --> ~H ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN )
82	80, 40, 81	sylancr 658	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN )
83	74	fconst 5593	. . . . . . . . . 10 |- ( NN X. { 0h } ) : NN --> { 0h }
84		ffn 5556	. . . . . . . . . 10 |- ( ( NN X. { 0h } ) : NN --> { 0h } -> ( NN X. { 0h } ) Fn NN )
85	83, 84	ax-mp 5	. . . . . . . . 9 |- ( NN X. { 0h } ) Fn NN
86		eqfnfv 5794	. . . . . . . . 9 |- ( ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN /\ ( NN X. { 0h } ) Fn NN ) -> ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) = ( NN X. { 0h } ) <-> A. k e. NN ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( NN X. { 0h } ) ` k ) ) )
87	82, 85, 86	sylancl 657	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) = ( NN X. { 0h } ) <-> A. k e. NN ( ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) ` k ) = ( ( NN X. { 0h } ) ` k ) ) )
88	78, 87	mpbird 232	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) = ( NN X. { 0h } ) )
89		vex 2973	. . . . . . . . . 10 |- x e. _V
90	89	hlimveci 24511	. . . . . . . . 9 |- ( f ~~>v x -> x e. ~H )
91	90	adantl 463	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ~H )
92		fveq2 5688	. . . . . . . . . 10 |- ( y = x -> ( T ` y ) = ( T ` x ) )
93		id 22	. . . . . . . . . 10 |- ( y = x -> y = x )
94	92, 93	oveq12d 6108	. . . . . . . . 9 |- ( y = x -> ( ( T ` y ) -h y ) = ( ( T ` x ) -h x ) )
95		ovex 6115	. . . . . . . . 9 |- ( ( T ` x ) -h x ) e. _V
96	94, 45, 95	fvmpt 5771	. . . . . . . 8 |- ( x e. ~H -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` x ) = ( ( T ` x ) -h x ) )
97	91, 96	syl 16	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` x ) = ( ( T ` x ) -h x ) )
98	36, 88, 97	3brtr3d 4318	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ( T ` x ) -h x ) )
99	73	a1i 11	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 0h e. ~H )
100		1zzd 10673	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 1 e. ZZ )
101		nnuz 10892	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
102	101	lmconst 18765	. . . . . . 7 |- ( ( ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) /\ 0h e. ~H /\ 1 e. ZZ ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
103	18, 99, 100, 102	syl3anc 1213	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
104	9, 98, 103	lmmo 18884	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( T ` x ) -h x ) = 0h )
105	19	ffvelrni 5839	. . . . . . 7 |- ( x e. ~H -> ( T ` x ) e. ~H )
106	91, 105	syl 16	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ~H )
107		hvsubeq0 24389	. . . . . 6 |- ( ( ( T ` x ) e. ~H /\ x e. ~H ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
108	106, 91, 107	syl2anc 656	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
109	104, 108	mpbid 210	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) = x )
110		fnfvelrn 5837	. . . . 5 |- ( ( T Fn ~H /\ x e. ~H ) -> ( T ` x ) e. ran T )
111	50, 91, 110	sylancr 658	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ran T )
112	109, 111	eqeltrrd 2516	. . 3 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
113	112	gen2 1597	. 2 |- A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
114		isch2 24545	. 2 |- ( ran T e. CH <-> ( ran T e. SH /\ A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T ) ) )
115	4, 113, 114	mpbir2an 906	1 |- ran T e. CH

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1362   = wceq 1364   e. wcel 1761   A.wral 2713   C_ wss 3325   {csn 3874   <.cop 3880   class class class wbr 4289   e. cmpt 4347   X. cxp 4834   ran crn 4837   |` cres 4838   o. ccom 4840   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   ^m cmap 7210   1c1 9279   NNcn 10318   ZZcz 10642   *Metcxmt 17701   MetOpencmopn 17706  TopOnctopon 18399   Cn ccn 18728   ~~> tclm 18730   Hauscha 18812   tX ctx 19033   NrmCVeccnv 23881   ~Hchil 24240   +h cva 24241   .h csm 24242   normhcno 24244   0hc0v 24245   -h cmv 24246   ~~>v chli 24248   SHcsh 24249   CHcch 24250   ConOpccop 24267   LinOpclo 24268   BndLinOpcbo 24269   HrmOpcho 24271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-dc 8611  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358  ax-hilex 24320  ax-hfvadd 24321  ax-hvcom 24322  ax-hvass 24323  ax-hv0cl 24324  ax-hvaddid 24325  ax-hfvmul 24326  ax-hvmulid 24327  ax-hvmulass 24328  ax-hvdistr1 24329  ax-hvdistr2 24330  ax-hvmul0 24331  ax-hfi 24400  ax-his1 24403  ax-his2 24404  ax-his3 24405  ax-his4 24406  ax-hcompl 24523

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-cn 18731  df-cnp 18732  df-lm 18733  df-t1 18818  df-haus 18819  df-cmp 18890  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-fcls 19414  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-cfil 20666  df-cau 20667  df-cmet 20668  df-grpo 23597  df-gid 23598  df-ginv 23599  df-gdiv 23600  df-ablo 23688  df-subgo 23708  df-vc 23843  df-nv 23889  df-va 23892  df-ba 23893  df-sm 23894  df-0v 23895  df-vs 23896  df-nmcv 23897  df-ims 23898  df-dip 24015  df-ssp 24039  df-lno 24063  df-nmoo 24064  df-blo 24065  df-0o 24066  df-ph 24132  df-cbn 24183  df-hlo 24206  df-hnorm 24289  df-hba 24290  df-hvsub 24292  df-hlim 24293  df-hcau 24294  df-sh 24528  df-ch 24543  df-oc 24574  df-ch0 24575  df-shs 24630  df-pjh 24717  df-h0op 25071  df-nmop 25162  df-cnop 25163  df-lnop 25164  df-bdop 25165  df-unop 25166  df-hmop 25167

This theorem is referenced by:  hmopidmpji  25475  hmopidmch  25476