Theorem hmopidmchi 25734

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 25525	. . . 4 |- ( T e. HrmOp -> T e. LinOp )
3	1, 2	ax-mp 5	. . 3 |- T e. LinOp
4	3	rnelshi 25642	. 2 |- ran T e. SH
5		eqid 2454	. . . . . . . 8 |- ( normh o. -h ) = ( normh o. -h )
6	5	hilxmet 24776	. . . . . . 7 |- ( normh o. -h ) e. ( *Met ` ~H )
7		eqid 2454	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
8	7	methaus 20237	. . . . . . 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 2454	. . . . . . . . . . . 12 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
11	10, 5	hhims 24753	. . . . . . . . . . . 12 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12	10, 11, 7	hhlm 24780	. . . . . . . . . . 11 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
13		resss 5245	. . . . . . . . . . 11 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
14	12, 13	eqsstri 3497	. . . . . . . . . 10 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	14	ssbri 4445	. . . . . . . . 9 |- ( f ~~>v x -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
16	15	adantl 466	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
17	7	mopntopon 20156	. . . . . . . . . 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 25552	. . . . . . . . . . . 12 |- T : ~H --> ~H
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T : ~H --> ~H )
21	20	feqmptd 5856	. . . . . . . . . 10 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T = ( y e. ~H |-> ( T ` y ) ) )
22		hmopbdoptHIL 25571	. . . . . . . . . . . . 13 |- ( T e. HrmOp -> T e. BndLinOp )
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 |- T e. BndLinOp
24		lnopcnbd 25619	. . . . . . . . . . . . 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 25487	. . . . . . . . . . 11 |- ConOp = ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
28	26, 27	eleqtri 2540	. . . . . . . . . 10 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
29	21, 28	syl6eqelr 2551	. . . . . . . . 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 19376	. . . . . . . . 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 24746	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. e. NrmCVec
32	10	hhvs 24751	. . . . . . . . . . 11 |- -h = ( -v ` <. <. +h , .h >. , normh >. )
33	11, 7, 32	vmcn 24273	. . . . . . . . . 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 19381	. . . . . . . 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 19051	. . . . . . 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 457	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ran T )
38	4	shssii 24794	. . . . . . . . . . . . . 14 |- ran T C_ ~H
39		fss 5678	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ ran T C_ ~H ) -> f : NN --> ~H )
40	37, 38, 39	sylancl 662	. . . . . . . . . . . . 13 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ~H )
41	40	ffvelrnda 5955	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ~H )
42		fveq2 5802	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( T ` y ) = ( T ` ( f ` k ) ) )
43		id 22	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> y = ( f ` k ) )
44	42, 43	oveq12d 6221	. . . . . . . . . . . . 13 |- ( y = ( f ` k ) -> ( ( T ` y ) -h y ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
45		eqid 2454	. . . . . . . . . . . . 13 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) = ( y e. ~H |-> ( ( T ` y ) -h y ) )
46		ovex 6228	. . . . . . . . . . . . 13 |- ( ( T ` ( f ` k ) ) -h ( f ` k ) ) e. _V
47	44, 45, 46	fvmpt 5886	. . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . 16 |- ( T : ~H --> ~H -> T Fn ~H )
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- T Fn ~H
51		fveq2 5802	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> ( T ` y ) = ( T ` ( T ` x ) ) )
52		id 22	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> y = ( T ` x ) )
53	51, 52	eqeq12d 2476	. . . . . . . . . . . . . . . 16 |- ( y = ( T ` x ) -> ( ( T ` y ) = y <-> ( T ` ( T ` x ) ) = ( T ` x ) ) )
54	53	ralrn 5958	. . . . . . . . . . . . . . 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 5803	. . . . . . . . . . . . . . 15 |- ( ( T o. T ) ` x ) = ( T ` x )
58	19, 19	hocoi 25347	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( T o. T ) ` x ) = ( T ` ( T ` x ) ) )
59	57, 58	syl5reqr 2510	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( T ` ( T ` x ) ) = ( T ` x ) )
60	55, 59	mprgbir 2904	. . . . . . . . . . . . 13 |- A. y e. ran T ( T ` y ) = y
61		ffvelrn 5953	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ k e. NN ) -> ( f ` k ) e. ran T )
62	61	adantlr 714	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ran T )
63	42, 43	eqeq12d 2476	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( ( T ` y ) = y <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
64	63	rspccv 3176	. . . . . . . . . . . . 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 2542	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) e. ~H )
67		hvsubeq0 24649	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 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 2495	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = 0h )
71		fvco3 5880	. . . . . . . . . . 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 714	. . . . . . . . . 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 24584	. . . . . . . . . . . . 13 |- 0h e. ~H
74	73	elexi 3088	. . . . . . . . . . . 12 |- 0h e. _V
75	74	fvconst2 6045	. . . . . . . . . . 11 |- ( k e. NN -> ( ( NN X. { 0h } ) ` k ) = 0h )
76	75	adantl 466	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( NN X. { 0h } ) ` k ) = 0h )
77	70, 72, 76	3eqtr4d 2505	. . . . . . . . 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 2830	. . . . . . . 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 6228	. . . . . . . . . . 11 |- ( ( T ` y ) -h y ) e. _V
80	79, 45	fnmpti 5650	. . . . . . . . . 10 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H
81		fnfco 5688	. . . . . . . . . 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 663	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN )
83	74	fconst 5707	. . . . . . . . . 10 |- ( NN X. { 0h } ) : NN --> { 0h }
84		ffn 5670	. . . . . . . . . 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 5909	. . . . . . . . 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 662	. . . . . . . 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 3081	. . . . . . . . . 10 |- x e. _V
90	89	hlimveci 24771	. . . . . . . . 9 |- ( f ~~>v x -> x e. ~H )
91	90	adantl 466	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ~H )
92		fveq2 5802	. . . . . . . . . 10 |- ( y = x -> ( T ` y ) = ( T ` x ) )
93		id 22	. . . . . . . . . 10 |- ( y = x -> y = x )
94	92, 93	oveq12d 6221	. . . . . . . . 9 |- ( y = x -> ( ( T ` y ) -h y ) = ( ( T ` x ) -h x ) )
95		ovex 6228	. . . . . . . . 9 |- ( ( T ` x ) -h x ) e. _V
96	94, 45, 95	fvmpt 5886	. . . . . . . 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 4432	. . . . . 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 10792	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 1 e. ZZ )
101		nnuz 11011	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
102	101	lmconst 19007	. . . . . . 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 1219	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
104	9, 98, 103	lmmo 19126	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( T ` x ) -h x ) = 0h )
105	19	ffvelrni 5954	. . . . . . 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 24649	. . . . . 6 |- ( ( ( T ` x ) e. ~H /\ x e. ~H ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
108	106, 91, 107	syl2anc 661	. . . . 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 5952	. . . . 5 |- ( ( T Fn ~H /\ x e. ~H ) -> ( T ` x ) e. ran T )
111	50, 91, 110	sylancr 663	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ran T )
112	109, 111	eqeltrrd 2543	. . 3 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
113	112	gen2 1593	. 2 |- A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
114		isch2 24805	. 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 911	1 |- ran T e. CH

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   = wceq 1370   e. wcel 1758   A.wral 2799   C_ wss 3439   {csn 3988   <.cop 3994   class class class wbr 4403   |-> cmpt 4461   X. cxp 4949   ran crn 4952   |` cres 4953   o. ccom 4955   Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   ^m cmap 7327   1c1 9398   NNcn 10437   ZZcz 10761   *Metcxmt 17936   MetOpencmopn 17941  TopOnctopon 18641   Cn ccn 18970   ~~> tclm 18972   Hauscha 19054   tX ctx 19275   NrmCVeccnv 24141   ~Hchil 24500   +h cva 24501   .h csm 24502   normhcno 24504   0hc0v 24505   -h cmv 24506   ~~>v chli 24508   SHcsh 24509   CHcch 24510   ConOpccop 24527   LinOpclo 24528   BndLinOpcbo 24529   HrmOpcho 24531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cc 8719  ax-dc 8730  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476  ax-mulf 9477  ax-hilex 24580  ax-hfvadd 24581  ax-hvcom 24582  ax-hvass 24583  ax-hv0cl 24584  ax-hvaddid 24585  ax-hfvmul 24586  ax-hvmulid 24587  ax-hvmulass 24588  ax-hvdistr1 24589  ax-hvdistr2 24590  ax-hvmul0 24591  ax-hfi 24660  ax-his1 24663  ax-his2 24664  ax-his3 24665  ax-his4 24666  ax-hcompl 24783

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7776  df-sup 7806  df-oi 7839  df-card 8224  df-acn 8227  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-ioo 11419  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-rlim 13089  df-sum 13286  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-starv 14376  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-unif 14384  df-hom 14385  df-cco 14386  df-rest 14484  df-topn 14485  df-0g 14503  df-gsum 14504  df-topgen 14505  df-pt 14506  df-prds 14509  df-xrs 14563  df-qtop 14568  df-imas 14569  df-xps 14571  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-submnd 15588  df-mulg 15671  df-cntz 15958  df-cmn 16404  df-psmet 17944  df-xmet 17945  df-met 17946  df-bl 17947  df-mopn 17948  df-fbas 17949  df-fg 17950  df-cnfld 17954  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-cn 18973  df-cnp 18974  df-lm 18975  df-t1 19060  df-haus 19061  df-cmp 19132  df-tx 19277  df-hmeo 19470  df-fil 19561  df-fm 19653  df-flim 19654  df-flf 19655  df-fcls 19656  df-xms 20037  df-ms 20038  df-tms 20039  df-cncf 20596  df-cfil 20908  df-cau 20909  df-cmet 20910  df-grpo 23857  df-gid 23858  df-ginv 23859  df-gdiv 23860  df-ablo 23948  df-subgo 23968  df-vc 24103  df-nv 24149  df-va 24152  df-ba 24153  df-sm 24154  df-0v 24155  df-vs 24156  df-nmcv 24157  df-ims 24158  df-dip 24275  df-ssp 24299  df-lno 24323  df-nmoo 24324  df-blo 24325  df-0o 24326  df-ph 24392  df-cbn 24443  df-hlo 24466  df-hnorm 24549  df-hba 24550  df-hvsub 24552  df-hlim 24553  df-hcau 24554  df-sh 24788  df-ch 24803  df-oc 24834  df-ch0 24835  df-shs 24890  df-pjh 24977  df-h0op 25331  df-nmop 25422  df-cnop 25423  df-lnop 25424  df-bdop 25425  df-unop 25426  df-hmop 25427

This theorem is referenced by:  hmopidmpji  25735  hmopidmch  25736