Theorem hmopidmchi 23607

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 23398	. . . 4 |- ( T e. HrmOp -> T e. LinOp )
3	1, 2	ax-mp 8	. . 3 |- T e. LinOp
4	3	rnelshi 23515	. 2 |- ran T e. SH
5		eqid 2404	. . . . . . . 8 |- ( normh o. -h ) = ( normh o. -h )
6	5	hilxmet 22650	. . . . . . 7 |- ( normh o. -h ) e. ( * Met ` ~H )
7		eqid 2404	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
8	7	methaus 18503	. . . . . . 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 2404	. . . . . . . . . . . 12 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
11	10, 5	hhims 22627	. . . . . . . . . . . 12 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12	10, 11, 7	hhlm 22654	. . . . . . . . . . 11 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
13		resss 5129	. . . . . . . . . . 11 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
14	12, 13	eqsstri 3338	. . . . . . . . . 10 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	14	ssbri 4214	. . . . . . . . 9 |- ( f ~~>v x -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
16	15	adantl 453	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
17	7	mopntopon 18422	. . . . . . . . . 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 23425	. . . . . . . . . . . 12 |- T : ~H --> ~H
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T : ~H --> ~H )
21	20	feqmptd 5738	. . . . . . . . . 10 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T = ( y e. ~H |-> ( T ` y ) ) )
22		hmopbdoptHIL 23444	. . . . . . . . . . . . 13 |- ( T e. HrmOp -> T e. BndLinOp )
23	1, 22	ax-mp 8	. . . . . . . . . . . 12 |- T e. BndLinOp
24		lnopcnbd 23492	. . . . . . . . . . . . 13 |- ( T e. LinOp -> ( T e. ConOp <-> T e. BndLinOp ) )
25	3, 24	ax-mp 8	. . . . . . . . . . . 12 |- ( T e. ConOp <-> T e. BndLinOp )
26	23, 25	mpbir 201	. . . . . . . . . . 11 |- T e. ConOp
27	5, 7	hhcno 23360	. . . . . . . . . . 11 |- ConOp = ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
28	26, 27	eleqtri 2476	. . . . . . . . . 10 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
29	21, 28	syl6eqelr 2493	. . . . . . . . 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 17646	. . . . . . . . 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 22620	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. e. NrmCVec
32	10	hhvs 22625	. . . . . . . . . . 11 |- -h = ( -v ` <. <. +h , .h >. , normh >. )
33	11, 7, 32	vmcn 22148	. . . . . . . . . 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 17651	. . . . . . . 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 17323	. . . . . . 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 444	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ran T )
38	4	shssii 22668	. . . . . . . . . . . . . 14 |- ran T C_ ~H
39		fss 5558	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ ran T C_ ~H ) -> f : NN --> ~H )
40	37, 38, 39	sylancl 644	. . . . . . . . . . . . 13 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ~H )
41	40	ffvelrnda 5829	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ~H )
42		fveq2 5687	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( T ` y ) = ( T ` ( f ` k ) ) )
43		id 20	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> y = ( f ` k ) )
44	42, 43	oveq12d 6058	. . . . . . . . . . . . 13 |- ( y = ( f ` k ) -> ( ( T ` y ) -h y ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
45		eqid 2404	. . . . . . . . . . . . 13 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) = ( y e. ~H |-> ( ( T ` y ) -h y ) )
46		ovex 6065	. . . . . . . . . . . . 13 |- ( ( T ` ( f ` k ) ) -h ( f ` k ) ) e. _V
47	44, 45, 46	fvmpt 5765	. . . . . . . . . . . 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 5550	. . . . . . . . . . . . . . . 16 |- ( T : ~H --> ~H -> T Fn ~H )
50	19, 49	ax-mp 8	. . . . . . . . . . . . . . 15 |- T Fn ~H
51		fveq2 5687	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> ( T ` y ) = ( T ` ( T ` x ) ) )
52		id 20	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> y = ( T ` x ) )
53	51, 52	eqeq12d 2418	. . . . . . . . . . . . . . . 16 |- ( y = ( T ` x ) -> ( ( T ` y ) = y <-> ( T ` ( T ` x ) ) = ( T ` x ) ) )
54	53	ralrn 5832	. . . . . . . . . . . . . . 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 8	. . . . . . . . . . . . . 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 5688	. . . . . . . . . . . . . . 15 |- ( ( T o. T ) ` x ) = ( T ` x )
58	19, 19	hocoi 23220	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( T o. T ) ` x ) = ( T ` ( T ` x ) ) )
59	57, 58	syl5reqr 2451	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( T ` ( T ` x ) ) = ( T ` x ) )
60	55, 59	mprgbir 2736	. . . . . . . . . . . . 13 |- A. y e. ran T ( T ` y ) = y
61		ffvelrn 5827	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ k e. NN ) -> ( f ` k ) e. ran T )
62	61	adantlr 696	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ran T )
63	42, 43	eqeq12d 2418	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( ( T ` y ) = y <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
64	63	rspccv 3009	. . . . . . . . . . . . 13 |- ( A. y e. ran T ( T ` y ) = y -> ( ( f ` k ) e. ran T -> ( T ` ( f ` k ) ) = ( f ` k ) ) )
65	60, 62, 64	mpsyl 61	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) = ( f ` k ) )
66	65, 41	eqeltrd 2478	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) e. ~H )
67		hvsubeq0 22523	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 224	. . . . . . . . . . 11 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( T ` ( f ` k ) ) -h ( f ` k ) ) = 0h )
70	48, 69	eqtrd 2436	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = 0h )
71		fvco3 5759	. . . . . . . . . . 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 696	. . . . . . . . . 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 22459	. . . . . . . . . . . . 13 |- 0h e. ~H
74	73	elexi 2925	. . . . . . . . . . . 12 |- 0h e. _V
75	74	fvconst2 5906	. . . . . . . . . . 11 |- ( k e. NN -> ( ( NN X. { 0h } ) ` k ) = 0h )
76	75	adantl 453	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( NN X. { 0h } ) ` k ) = 0h )
77	70, 72, 76	3eqtr4d 2446	. . . . . . . . 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 2749	. . . . . . . 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 6065	. . . . . . . . . . 11 |- ( ( T ` y ) -h y ) e. _V
80	79, 45	fnmpti 5532	. . . . . . . . . 10 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H
81		fnfco 5568	. . . . . . . . . 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 645	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN )
83	74	fconst 5588	. . . . . . . . . 10 |- ( NN X. { 0h } ) : NN --> { 0h }
84		ffn 5550	. . . . . . . . . 10 |- ( ( NN X. { 0h } ) : NN --> { 0h } -> ( NN X. { 0h } ) Fn NN )
85	83, 84	ax-mp 8	. . . . . . . . 9 |- ( NN X. { 0h } ) Fn NN
86		eqfnfv 5786	. . . . . . . . 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 644	. . . . . . . 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 224	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) = ( NN X. { 0h } ) )
89		vex 2919	. . . . . . . . . 10 |- x e. _V
90	89	hlimveci 22645	. . . . . . . . 9 |- ( f ~~>v x -> x e. ~H )
91	90	adantl 453	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ~H )
92		fveq2 5687	. . . . . . . . . 10 |- ( y = x -> ( T ` y ) = ( T ` x ) )
93		id 20	. . . . . . . . . 10 |- ( y = x -> y = x )
94	92, 93	oveq12d 6058	. . . . . . . . 9 |- ( y = x -> ( ( T ` y ) -h y ) = ( ( T ` x ) -h x ) )
95		ovex 6065	. . . . . . . . 9 |- ( ( T ` x ) -h x ) e. _V
96	94, 45, 95	fvmpt 5765	. . . . . . . 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 4201	. . . . . 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		1z 10267	. . . . . . . 8 |- 1 e. ZZ
101	100	a1i 11	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 1 e. ZZ )
102		nnuz 10477	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
103	102	lmconst 17279	. . . . . . 7 |- ( ( ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) /\ 0h e. ~H /\ 1 e. ZZ ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
104	18, 99, 101, 103	syl3anc 1184	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
105	9, 98, 104	lmmo 17398	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( T ` x ) -h x ) = 0h )
106	19	ffvelrni 5828	. . . . . . 7 |- ( x e. ~H -> ( T ` x ) e. ~H )
107	91, 106	syl 16	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ~H )
108		hvsubeq0 22523	. . . . . 6 |- ( ( ( T ` x ) e. ~H /\ x e. ~H ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
109	107, 91, 108	syl2anc 643	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
110	105, 109	mpbid 202	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) = x )
111		fnfvelrn 5826	. . . . 5 |- ( ( T Fn ~H /\ x e. ~H ) -> ( T ` x ) e. ran T )
112	50, 91, 111	sylancr 645	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ran T )
113	110, 112	eqeltrrd 2479	. . 3 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
114	113	gen2 1553	. 2 |- A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
115		isch2 22679	. 2 |- ( ran T e. CH <-> ( ran T e. SH /\ A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T ) ) )
116	4, 114, 115	mpbir2an 887	1 |- ran T e. CH

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   A.wal 1546   = wceq 1649   e. wcel 1721   A.wral 2666   C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172   e. cmpt 4226   X. cxp 4835   ran crn 4838   |` cres 4839   o. ccom 4841   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   ^m cmap 6977   1c1 8947   NNcn 9956   ZZcz 10238   * Metcxmt 16641   MetOpencmopn 16646  TopOnctopon 16914   Cn ccn 17242   ~~> tclm 17244   Hauscha 17326   tX ctx 17545   NrmCVeccnv 22016   ~Hchil 22375   +h cva 22376   .h csm 22377   normhcno 22379   0hc0v 22380   -h cmv 22381   ~~>v chli 22383   SHcsh 22384   CHcch 22385   ConOpccop 22402   LinOpclo 22403   BndLinOpcbo 22404   HrmOpcho 22406

This theorem is referenced by:  hmopidmpji  23608  hmopidmch  23609

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-dc 8282  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540  ax-hcompl 22657

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-lm 17247  df-t1 17332  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-fcls 17926  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-cfil 19161  df-cau 19162  df-cmet 19163  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-dip 22150  df-ssp 22174  df-lno 22198  df-nmoo 22199  df-blo 22200  df-0o 22201  df-ph 22267  df-cbn 22318  df-hlo 22341  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-hlim 22428  df-hcau 22429  df-sh 22662  df-ch 22677  df-oc 22707  df-ch0 22708  df-shs 22763  df-pjh 22850  df-h0op 23204  df-nmop 23295  df-cnop 23296  df-lnop 23297  df-bdop 23298  df-unop 23299  df-hmop 23300