Theorem hmopidmchi 27885

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 27676	. . . 4 |- ( T e. HrmOp -> T e. LinOp )
3	1, 2	ax-mp 5	. . 3 |- T e. LinOp
4	3	rnelshi 27793	. 2 |- ran T e. SH
5		eqid 2471	. . . . . . . 8 |- ( normh o. -h ) = ( normh o. -h )
6	5	hilxmet 26929	. . . . . . 7 |- ( normh o. -h ) e. ( *Met ` ~H )
7		eqid 2471	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
8	7	methaus 21613	. . . . . . 7 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. Haus )
9	6, 8	mp1i 13	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( MetOpen ` ( normh o. -h ) ) e. Haus )
10		eqid 2471	. . . . . . . . . . . 12 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
11	10, 5	hhims 26906	. . . . . . . . . . . 12 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12	10, 11, 7	hhlm 26933	. . . . . . . . . . 11 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
13		resss 5134	. . . . . . . . . . 11 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
14	12, 13	eqsstri 3448	. . . . . . . . . 10 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	14	ssbri 4438	. . . . . . . . 9 |- ( f ~~>v x -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
16	15	adantl 473	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
17	7	mopntopon 21532	. . . . . . . . . 10 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
18	6, 17	mp1i 13	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
19	3	lnopfi 27703	. . . . . . . . . . . 12 |- T : ~H --> ~H
20	19	a1i 11	. . . . . . . . . . 11 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T : ~H --> ~H )
21	20	feqmptd 5932	. . . . . . . . . 10 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> T = ( y e. ~H |-> ( T ` y ) ) )
22		hmopbdoptHIL 27722	. . . . . . . . . . . . 13 |- ( T e. HrmOp -> T e. BndLinOp )
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 |- T e. BndLinOp
24		lnopcnbd 27770	. . . . . . . . . . . . 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 214	. . . . . . . . . . 11 |- T e. ConOp
27	5, 7	hhcno 27638	. . . . . . . . . . 11 |- ConOp = ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
28	26, 27	eleqtri 2547	. . . . . . . . . 10 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) )
29	21, 28	syl6eqelr 2558	. . . . . . . . 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 20753	. . . . . . . . 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 26899	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. e. NrmCVec
32	10	hhvs 26904	. . . . . . . . . . 11 |- -h = ( -v ` <. <. +h , .h >. , normh >. )
33	11, 7, 32	vmcn 26416	. . . . . . . . . 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 13	. . . . . . . . 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 20758	. . . . . . . 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 20398	. . . . . . 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 464	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ran T )
38	4	shssii 26947	. . . . . . . . . . . . . 14 |- ran T C_ ~H
39		fss 5749	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ ran T C_ ~H ) -> f : NN --> ~H )
40	37, 38, 39	sylancl 675	. . . . . . . . . . . . 13 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> f : NN --> ~H )
41	40	ffvelrnda 6037	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ~H )
42		fveq2 5879	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( T ` y ) = ( T ` ( f ` k ) ) )
43		id 22	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> y = ( f ` k ) )
44	42, 43	oveq12d 6326	. . . . . . . . . . . . 13 |- ( y = ( f ` k ) -> ( ( T ` y ) -h y ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
45		eqid 2471	. . . . . . . . . . . . 13 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) = ( y e. ~H |-> ( ( T ` y ) -h y ) )
46		ovex 6336	. . . . . . . . . . . . 13 |- ( ( T ` ( f ` k ) ) -h ( f ` k ) ) e. _V
47	44, 45, 46	fvmpt 5963	. . . . . . . . . . . 12 |- ( ( f ` k ) e. ~H -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = ( ( T ` ( f ` k ) ) -h ( f ` k ) ) )
48	41, 47	syl 17	. . . . . . . . . . 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 5739	. . . . . . . . . . . . . . . 16 |- ( T : ~H --> ~H -> T Fn ~H )
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- T Fn ~H
51		fveq2 5879	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> ( T ` y ) = ( T ` ( T ` x ) ) )
52		id 22	. . . . . . . . . . . . . . . . 17 |- ( y = ( T ` x ) -> y = ( T ` x ) )
53	51, 52	eqeq12d 2486	. . . . . . . . . . . . . . . 16 |- ( y = ( T ` x ) -> ( ( T ` y ) = y <-> ( T ` ( T ` x ) ) = ( T ` x ) ) )
54	53	ralrn 6040	. . . . . . . . . . . . . . 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 5880	. . . . . . . . . . . . . . 15 |- ( ( T o. T ) ` x ) = ( T ` x )
58	19, 19	hocoi 27498	. . . . . . . . . . . . . . 15 |- ( x e. ~H -> ( ( T o. T ) ` x ) = ( T ` ( T ` x ) ) )
59	57, 58	syl5reqr 2520	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( T ` ( T ` x ) ) = ( T ` x ) )
60	55, 59	mprgbir 2771	. . . . . . . . . . . . 13 |- A. y e. ran T ( T ` y ) = y
61		ffvelrn 6035	. . . . . . . . . . . . . 14 |- ( ( f : NN --> ran T /\ k e. NN ) -> ( f ` k ) e. ran T )
62	61	adantlr 729	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( f ` k ) e. ran T )
63	42, 43	eqeq12d 2486	. . . . . . . . . . . . . 14 |- ( y = ( f ` k ) -> ( ( T ` y ) = y <-> ( T ` ( f ` k ) ) = ( f ` k ) ) )
64	63	rspccv 3133	. . . . . . . . . . . . 13 |- ( A. y e. ran T ( T ` y ) = y -> ( ( f ` k ) e. ran T -> ( T ` ( f ` k ) ) = ( f ` k ) ) )
65	60, 62, 64	mpsyl 64	. . . . . . . . . . . 12 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) = ( f ` k ) )
66	65, 41	eqeltrd 2549	. . . . . . . . . . . . 13 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( T ` ( f ` k ) ) e. ~H )
67		hvsubeq0 26802	. . . . . . . . . . . . 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 673	. . . . . . . . . . . 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 240	. . . . . . . . . . 11 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( T ` ( f ` k ) ) -h ( f ` k ) ) = 0h )
70	48, 69	eqtrd 2505	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` ( f ` k ) ) = 0h )
71		fvco3 5957	. . . . . . . . . . 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 729	. . . . . . . . . 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 26737	. . . . . . . . . . . . 13 |- 0h e. ~H
74	73	elexi 3041	. . . . . . . . . . . 12 |- 0h e. _V
75	74	fvconst2 6136	. . . . . . . . . . 11 |- ( k e. NN -> ( ( NN X. { 0h } ) ` k ) = 0h )
76	75	adantl 473	. . . . . . . . . 10 |- ( ( ( f : NN --> ran T /\ f ~~>v x ) /\ k e. NN ) -> ( ( NN X. { 0h } ) ` k ) = 0h )
77	70, 72, 76	3eqtr4d 2515	. . . . . . . . 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 2809	. . . . . . . 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 6336	. . . . . . . . . . 11 |- ( ( T ` y ) -h y ) e. _V
80	79, 45	fnmpti 5716	. . . . . . . . . 10 |- ( y e. ~H |-> ( ( T ` y ) -h y ) ) Fn ~H
81		fnfco 5760	. . . . . . . . . 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 676	. . . . . . . . 9 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) Fn NN )
83	74	fconst 5782	. . . . . . . . . 10 |- ( NN X. { 0h } ) : NN --> { 0h }
84		ffn 5739	. . . . . . . . . 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 5991	. . . . . . . . 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 675	. . . . . . . 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 240	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) o. f ) = ( NN X. { 0h } ) )
89		vex 3034	. . . . . . . . . 10 |- x e. _V
90	89	hlimveci 26924	. . . . . . . . 9 |- ( f ~~>v x -> x e. ~H )
91	90	adantl 473	. . . . . . . 8 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ~H )
92		fveq2 5879	. . . . . . . . . 10 |- ( y = x -> ( T ` y ) = ( T ` x ) )
93		id 22	. . . . . . . . . 10 |- ( y = x -> y = x )
94	92, 93	oveq12d 6326	. . . . . . . . 9 |- ( y = x -> ( ( T ` y ) -h y ) = ( ( T ` x ) -h x ) )
95		ovex 6336	. . . . . . . . 9 |- ( ( T ` x ) -h x ) e. _V
96	94, 45, 95	fvmpt 5963	. . . . . . . 8 |- ( x e. ~H -> ( ( y e. ~H |-> ( ( T ` y ) -h y ) ) ` x ) = ( ( T ` x ) -h x ) )
97	91, 96	syl 17	. . . . . . 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 4425	. . . . . 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 10992	. . . . . . 7 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> 1 e. ZZ )
101		nnuz 11218	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
102	101	lmconst 20354	. . . . . . 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 1292	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( NN X. { 0h } ) ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) 0h )
104	9, 98, 103	lmmo 20473	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( T ` x ) -h x ) = 0h )
105	19	ffvelrni 6036	. . . . . . 7 |- ( x e. ~H -> ( T ` x ) e. ~H )
106	91, 105	syl 17	. . . . . 6 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ~H )
107		hvsubeq0 26802	. . . . . 6 |- ( ( ( T ` x ) e. ~H /\ x e. ~H ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
108	106, 91, 107	syl2anc 673	. . . . 5 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( ( ( T ` x ) -h x ) = 0h <-> ( T ` x ) = x ) )
109	104, 108	mpbid 215	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) = x )
110		fnfvelrn 6034	. . . . 5 |- ( ( T Fn ~H /\ x e. ~H ) -> ( T ` x ) e. ran T )
111	50, 91, 110	sylancr 676	. . . 4 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> ( T ` x ) e. ran T )
112	109, 111	eqeltrrd 2550	. . 3 |- ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
113	112	gen2 1678	. 2 |- A. f A. x ( ( f : NN --> ran T /\ f ~~>v x ) -> x e. ran T )
114		isch2 26957	. 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 934	1 |- ran T e. CH

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   A.wal 1450   = wceq 1452   e. wcel 1904   A.wral 2756   C_ wss 3390   {csn 3959   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   ran crn 4840   |` cres 4841   o. ccom 4843   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   ^m cmap 7490   1c1 9558   NNcn 10631   ZZcz 10961   *Metcxmt 19032   MetOpencmopn 19037  TopOnctopon 19995   Cn ccn 20317   ~~> tclm 20319   Hauscha 20401   tX ctx 20652   NrmCVeccnv 26284   ~Hchil 26653   +h cva 26654   .h csm 26655   normhcno 26657   0hc0v 26658   -h cmv 26659   ~~>v chli 26661   SHcsh 26662   CHcch 26663   ConOpccop 26680   LinOpclo 26681   BndLinOpcbo 26682   HrmOpcho 26684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-dc 8894  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637  ax-hilex 26733  ax-hfvadd 26734  ax-hvcom 26735  ax-hvass 26736  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvdistr1 26742  ax-hvdistr2 26743  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his2 26817  ax-his3 26818  ax-his4 26819  ax-hcompl 26936

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-lm 20322  df-t1 20407  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-fcls 21034  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-cfil 22303  df-cau 22304  df-cmet 22305  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-vs 26299  df-nmcv 26300  df-ims 26301  df-dip 26418  df-ssp 26442  df-lno 26466  df-nmoo 26467  df-blo 26468  df-0o 26469  df-ph 26535  df-cbn 26586  df-hlo 26619  df-hnorm 26702  df-hba 26703  df-hvsub 26705  df-hlim 26706  df-hcau 26707  df-sh 26941  df-ch 26955  df-oc 26986  df-ch0 26987  df-shs 27042  df-pjh 27129  df-h0op 27482  df-nmop 27573  df-cnop 27574  df-lnop 27575  df-bdop 27576  df-unop 27577  df-hmop 27578

This theorem is referenced by:  hmopidmpji  27886  hmopidmch  27887