Theorem hmopidmchi 26893

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 26684	. . . 4 |- ( T e. HrmOp -> T e. LinOp )
3	1, 2	ax-mp 5	. . 3 |- T e. LinOp
4	3	rnelshi 26801	. 2 |- ran T e. SH
5		eqid 2467	. . . . . . . 8 |- ( normh o. -h ) = ( normh o. -h )
6	5	hilxmet 25935	. . . . . . 7 |- ( normh o. -h ) e. ( *Met ` ~H )
7		eqid 2467	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
8	7	methaus 20891	. . . . . . 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 2467	. . . . . . . . . . . 12 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
11	10, 5	hhims 25912	. . . . . . . . . . . 12 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12	10, 11, 7	hhlm 25939	. . . . . . . . . . 11 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
13		resss 5303	. . . . . . . . . . 11 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
14	12, 13	eqsstri 3539	. . . . . . . . . 10 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	14	ssbri 4495	. . . . . . . . 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 20810	. . . . . . . . . 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 26711	. . . . . . . . . . . 12 |- T : ~H --> ~H
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T : ~H --> ~H )
21	20	feqmptd 5927	. . . . . . . . . 10 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T = ( y e. ~H |-> ( T ` y ) ) )
22		hmopbdoptHIL 26730	. . . . . . . . . . . . 13 |- ( T e. HrmOp -> T e. BndLinOp )
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 |- T e. BndLinOp
24		lnopcnbd 26778	. . . . . . . . . . . . 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 26646	. . . . . . . . . . 11 |- ConOp = ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
28	26, 27	eleqtri 2553	. . . . . . . . . 10 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
29	21, 28	syl6eqelr 2564	. . . . . . . . 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 20030	. . . . . . . . 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 25905	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. e. NrmCVec
32	10	hhvs 25910	. . . . . . . . . . 11 |- -h = ( -v ` <. <. +h , .h >. , normh >. )
33	11, 7, 32	vmcn 25432	. . . . . . . . . 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 20035	. . . . . . . 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 19674	. . . . . . 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 25953	. . . . . . . . . . . . . 14 |- ran T C_ ~H
39		fss 5745	. . . . . . . . . . . . . 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 6032	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ~H )
42		fveq2 5872	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( T ` y ) = ( T ` ( f ` k ) ) )
43		id 22	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> y = ( f ` k ) )
44	42, 43	oveq12d 6313	. . . . . . . . . . . . 13 |- ( y = ( f ` k ) -> ( ( T ` y ) -h y ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
45		eqid 2467	. . . . . . . . . . . . 13 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) = ( y e. ~H |-> ( ( T ` y ) -h y ) )
46		ovex 6320	. . . . . . . . . . . . 13 |- ( ( T ` ( f ` k ) ) -h ( f ` k ) ) e. _V
47	44, 45, 46	fvmpt 5957	. . . . . . . . . . . 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 5737	. . . . . . . . . . . . . . . 16 |- ( T : ~H --> ~H -> T Fn ~H )
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- T Fn ~H
51		fveq2 5872	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> ( T ` y ) = ( T ` ( T ` x ) ) )
52		id 22	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> y = ( T ` x ) )
53	51, 52	eqeq12d 2489	. . . . . . . . . . . . . . . 16 |- ( y = ( T ` x ) -> ( ( T ` y ) = y <-> ( T ` ( T ` x ) ) = ( T ` x ) ) )
54	53	ralrn 6035	. . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . 15 |- ( ( T o. T ) ` x ) = ( T ` x )
58	19, 19	hocoi 26506	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( T o. T ) ` x ) = ( T ` ( T ` x ) ) )
59	57, 58	syl5reqr 2523	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( T ` ( T ` x ) ) = ( T ` x ) )
60	55, 59	mprgbir 2831	. . . . . . . . . . . . 13 |- A. y e. ran T ( T ` y ) = y
61		ffvelrn 6030	. . . . . . . . . . . . . 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 2489	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( ( T ` y ) = y <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
64	63	rspccv 3216	. . . . . . . . . . . . 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 2555	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) e. ~H )
67		hvsubeq0 25808	. . . . . . . . . . . . 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 2508	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = 0h )
71		fvco3 5951	. . . . . . . . . . 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 25743	. . . . . . . . . . . . 13 |- 0h e. ~H
74	73	elexi 3128	. . . . . . . . . . . 12 |- 0h e. _V
75	74	fvconst2 6127	. . . . . . . . . . 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 2518	. . . . . . . . 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 2881	. . . . . . . 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 6320	. . . . . . . . . . 11 |- ( ( T ` y ) -h y ) e. _V
80	79, 45	fnmpti 5715	. . . . . . . . . 10 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H
81		fnfco 5756	. . . . . . . . . 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 5777	. . . . . . . . . 10 |- ( NN X. { 0h } ) : NN --> { 0h }
84		ffn 5737	. . . . . . . . . 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 5982	. . . . . . . . 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 3121	. . . . . . . . . 10 |- x e. _V
90	89	hlimveci 25930	. . . . . . . . 9 |- ( f ~~>v x -> x e. ~H )
91	90	adantl 466	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ~H )
92		fveq2 5872	. . . . . . . . . 10 |- ( y = x -> ( T ` y ) = ( T ` x ) )
93		id 22	. . . . . . . . . 10 |- ( y = x -> y = x )
94	92, 93	oveq12d 6313	. . . . . . . . 9 |- ( y = x -> ( ( T ` y ) -h y ) = ( ( T ` x ) -h x ) )
95		ovex 6320	. . . . . . . . 9 |- ( ( T ` x ) -h x ) e. _V
96	94, 45, 95	fvmpt 5957	. . . . . . . 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 4482	. . . . . 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 10907	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 1 e. ZZ )
101		nnuz 11129	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
102	101	lmconst 19630	. . . . . . 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 1228	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
104	9, 98, 103	lmmo 19749	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( T ` x ) -h x ) = 0h )
105	19	ffvelrni 6031	. . . . . . 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 25808	. . . . . 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 6029	. . . . 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 2556	. . 3 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
113	112	gen2 1602	. 2 |- A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
114		isch2 25964	. 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 918	1 |- ran T e. CH

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   = wceq 1379   e. wcel 1767   A.wral 2817   C_ wss 3481   {csn 4033   <.cop 4039   class class class wbr 4453   |-> cmpt 4511   X. cxp 5003   ran crn 5006   |` cres 5007   o. ccom 5009   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   ^m cmap 7432   1c1 9505   NNcn 10548   ZZcz 10876   *Metcxmt 18273   MetOpencmopn 18278  TopOnctopon 19264   Cn ccn 19593   ~~> tclm 19595   Hauscha 19677   tX ctx 19929   NrmCVeccnv 25300   ~Hchil 25659   +h cva 25660   .h csm 25661   normhcno 25663   0hc0v 25664   -h cmv 25665   ~~>v chli 25667   SHcsh 25668   CHcch 25669   ConOpccop 25686   LinOpclo 25687   BndLinOpcbo 25688   HrmOpcho 25690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cc 8827  ax-dc 8838  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584  ax-hilex 25739  ax-hfvadd 25740  ax-hvcom 25741  ax-hvass 25742  ax-hv0cl 25743  ax-hvaddid 25744  ax-hfvmul 25745  ax-hvmulid 25746  ax-hvmulass 25747  ax-hvdistr1 25748  ax-hvdistr2 25749  ax-hvmul0 25750  ax-hfi 25819  ax-his1 25822  ax-his2 25823  ax-his3 25824  ax-his4 25825  ax-hcompl 25942

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-cn 19596  df-cnp 19597  df-lm 19598  df-t1 19683  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-fcls 20310  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-cfil 21562  df-cau 21563  df-cmet 21564  df-grpo 25016  df-gid 25017  df-ginv 25018  df-gdiv 25019  df-ablo 25107  df-subgo 25127  df-vc 25262  df-nv 25308  df-va 25311  df-ba 25312  df-sm 25313  df-0v 25314  df-vs 25315  df-nmcv 25316  df-ims 25317  df-dip 25434  df-ssp 25458  df-lno 25482  df-nmoo 25483  df-blo 25484  df-0o 25485  df-ph 25551  df-cbn 25602  df-hlo 25625  df-hnorm 25708  df-hba 25709  df-hvsub 25711  df-hlim 25712  df-hcau 25713  df-sh 25947  df-ch 25962  df-oc 25993  df-ch0 25994  df-shs 26049  df-pjh 26136  df-h0op 26490  df-nmop 26581  df-cnop 26582  df-lnop 26583  df-bdop 26584  df-unop 26585  df-hmop 26586

This theorem is referenced by:  hmopidmpji  26894  hmopidmch  26895