Theorem stoweidlem62 31378

Description: This theorem proves the Stone Weierstrass theorem for the non-trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem62.1	|- F/_ t F
stoweidlem62.2	|- F/ f ph
stoweidlem62.3	|- F/ t ph
stoweidlem62.4	|- H = ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
stoweidlem62.5	|- K = ( topGen ` ran (,) )
stoweidlem62.6	|- T = U. J
stoweidlem62.7	|- ( ph -> J e. Comp )
stoweidlem62.8	|- C = ( J Cn K )
stoweidlem62.9	|- ( ph -> A C_ C )
stoweidlem62.10	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem62.11	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweidlem62.12	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweidlem62.13	|- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
stoweidlem62.14	|- ( ph -> F e. C )
stoweidlem62.15	|- ( ph -> E e. RR+ )
stoweidlem62.16	|- ( ph -> T =/= (/) )
stoweidlem62.17	|- ( ph -> E < ( 1 / 3 ) )

Assertion

Ref	Expression
stoweidlem62	|- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Distinct variable groups:   f, g, t, A   f, q, r, x, t, A   f, E, g, t   f, F, g   f, H, g   f, J, r, t   T, f, g, t   ph, f, g   E, q, r, x   H, q, r, x   T, q, r, x   ph, q, r, x   t, K   x, F

Allowed substitution hints:   ph( t)   C( x, t, f, g, r, q)   F( t, r, q)   H( t)   J( x, g, q)   K( x, f, g, r, q)

Proof of Theorem stoweidlem62

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem62.4	. . . . 5 |- H = ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
2		nfmpt1 4536	. . . . 5 |- F/_ t ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
3	1, 2	nfcxfr 2627	. . . 4 |- F/_ t H
4		stoweidlem62.3	. . . 4 |- F/ t ph
5		stoweidlem62.5	. . . 4 |- K = ( topGen ` ran (,) )
6		stoweidlem62.7	. . . 4 |- ( ph -> J e. Comp )
7		stoweidlem62.6	. . . 4 |- T = U. J
8		stoweidlem62.16	. . . 4 |- ( ph -> T =/= (/) )
9		stoweidlem62.8	. . . 4 |- C = ( J Cn K )
10		stoweidlem62.9	. . . 4 |- ( ph -> A C_ C )
11		eleq1 2539	. . . . . . 7 |- ( g = h -> ( g e. A <-> h e. A ) )
12	11	3anbi3d 1305	. . . . . 6 |- ( g = h -> ( ( ph /\ f e. A /\ g e. A ) <-> ( ph /\ f e. A /\ h e. A ) ) )
13		fveq1 5864	. . . . . . . . 9 |- ( g = h -> ( g ` t ) = ( h ` t ) )
14	13	oveq2d 6299	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) + ( g ` t ) ) = ( ( f ` t ) + ( h ` t ) ) )
15	14	mpteq2dv 4534	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) )
16	15	eleq1d 2536	. . . . . 6 |- ( g = h -> ( ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A <-> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A ) )
17	12, 16	imbi12d 320	. . . . 5 |- ( g = h -> ( ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A ) <-> ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A ) ) )
18		stoweidlem62.10	. . . . 5 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
19	17, 18	chvarv 1983	. . . 4 |- ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A )
20	13	oveq2d 6299	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) x. ( g ` t ) ) = ( ( f ` t ) x. ( h ` t ) ) )
21	20	mpteq2dv 4534	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) )
22	21	eleq1d 2536	. . . . . 6 |- ( g = h -> ( ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A <-> ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) e. A ) )
23	12, 22	imbi12d 320	. . . . 5 |- ( g = h -> ( ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A ) <-> ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) e. A ) ) )
24		stoweidlem62.11	. . . . 5 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
25	23, 24	chvarv 1983	. . . 4 |- ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) e. A )
26		stoweidlem62.12	. . . 4 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
27		stoweidlem62.13	. . . 4 |- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
28		stoweidlem62.1	. . . . . 6 |- F/_ t F
29	28	nfrn 5244	. . . . . . 7 |- F/_ t ran F
30		nfcv 2629	. . . . . . 7 |- F/_ t RR
31		nfcv 2629	. . . . . . 7 |- F/_ t `' <
32	29, 30, 31	nfsup 7910	. . . . . 6 |- F/_ t sup ( ran F , RR , `' < )
33		eqid 2467	. . . . . 6 |- ( T X. { -u sup ( ran F , RR , `' < ) } ) = ( T X. { -u sup ( ran F , RR , `' < ) } )
34		cmptop 19677	. . . . . . 7 |- ( J e. Comp -> J e. Top )
35	6, 34	syl 16	. . . . . 6 |- ( ph -> J e. Top )
36		stoweidlem62.14	. . . . . 6 |- ( ph -> F e. C )
37	36, 9	syl6eleq 2565	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
38	28, 4, 7, 5, 6, 37, 8	stoweidlem29 31345	. . . . . . 7 |- ( ph -> ( sup ( ran F , RR , `' < ) e. ran F /\ sup ( ran F , RR , `' < ) e. RR /\ A. t e. T sup ( ran F , RR , `' < ) <_ ( F ` t ) ) )
39	38	simp2d 1009	. . . . . 6 |- ( ph -> sup ( ran F , RR , `' < ) e. RR )
40	28, 32, 4, 7, 33, 5, 35, 9, 36, 39	stoweidlem47 31363	. . . . 5 |- ( ph -> ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) e. C )
41	1, 40	syl5eqel 2559	. . . 4 |- ( ph -> H e. C )
42	38	simp3d 1010	. . . . . . . . 9 |- ( ph -> A. t e. T sup ( ran F , RR , `' < ) <_ ( F ` t ) )
43	42	r19.21bi 2833	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> sup ( ran F , RR , `' < ) <_ ( F ` t ) )
44	5, 7, 9, 36	fcnre 30994	. . . . . . . . . 10 |- ( ph -> F : T --> RR )
45	44	fnvinran 30983	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
46	39	adantr 465	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sup ( ran F , RR , `' < ) e. RR )
47	45, 46	subge0d 10141	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( 0 <_ ( ( F ` t ) - sup ( ran F , RR , `' < ) ) <-> sup ( ran F , RR , `' < ) <_ ( F ` t ) ) )
48	43, 47	mpbird 232	. . . . . . 7 |- ( ( ph /\ t e. T ) -> 0 <_ ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
49		simpr 461	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
50	45, 46	resubcld 9986	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) e. RR )
51	1	fvmpt2 5956	. . . . . . . 8 |- ( ( t e. T /\ ( ( F ` t ) - sup ( ran F , RR , `' < ) ) e. RR ) -> ( H ` t ) = ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
52	49, 50, 51	syl2anc 661	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) = ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
53	48, 52	breqtrrd 4473	. . . . . 6 |- ( ( ph /\ t e. T ) -> 0 <_ ( H ` t ) )
54	53	ex 434	. . . . 5 |- ( ph -> ( t e. T -> 0 <_ ( H ` t ) ) )
55	4, 54	ralrimi 2864	. . . 4 |- ( ph -> A. t e. T 0 <_ ( H ` t ) )
56		stoweidlem62.15	. . . . 5 |- ( ph -> E e. RR+ )
57	56	rphalfcld 11267	. . . 4 |- ( ph -> ( E / 2 ) e. RR+ )
58	56	rpred 11255	. . . . . 6 |- ( ph -> E e. RR )
59	58	rehalfcld 10784	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
60		3re 10608	. . . . . . 7 |- 3 e. RR
61		3ne0 10629	. . . . . . 7 |- 3 =/= 0
62	60, 61	rereccli 10308	. . . . . 6 |- ( 1 / 3 ) e. RR
63	62	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. RR )
64		rphalflt 11245	. . . . . 6 |- ( E e. RR+ -> ( E / 2 ) < E )
65	56, 64	syl 16	. . . . 5 |- ( ph -> ( E / 2 ) < E )
66		stoweidlem62.17	. . . . 5 |- ( ph -> E < ( 1 / 3 ) )
67	59, 58, 63, 65, 66	lttrd 9741	. . . 4 |- ( ph -> ( E / 2 ) < ( 1 / 3 ) )
68	3, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67	stoweidlem61 31377	. . 3 |- ( ph -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
69		nfra1 2845	. . . . . . 7 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) )
70	4, 69	nfan 1875	. . . . . 6 |- F/ t ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
71		rsp 2830	. . . . . . 7 |- ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> ( t e. T -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) ) )
72	56	rpcnd 11257	. . . . . . . . . 10 |- ( ph -> E e. CC )
73		2cnd 10607	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
74		2ne0 10627	. . . . . . . . . . 11 |- 2 =/= 0
75	74	a1i 11	. . . . . . . . . 10 |- ( ph -> 2 =/= 0 )
76	72, 73, 75	divcan2d 10321	. . . . . . . . 9 |- ( ph -> ( 2 x. ( E / 2 ) ) = E )
77	76	breq2d 4459	. . . . . . . 8 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) <-> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
78	77	biimpd 207	. . . . . . 7 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
79	71, 78	sylan9r 658	. . . . . 6 |- ( ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) ) -> ( t e. T -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
80	70, 79	ralrimi 2864	. . . . 5 |- ( ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) ) -> A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
81	80	ex 434	. . . 4 |- ( ph -> ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
82	81	reximdv 2937	. . 3 |- ( ph -> ( E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
83	68, 82	mpd 15	. 2 |- ( ph -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
84		nfmpt1 4536	. . 3 |- F/_ t ( t e. T |-> ( ( h ` t ) + sup ( ran F , RR , `' < ) ) )
85		nfcv 2629	. . 3 |- F/_ t h
86		nfv 1683	. . . . 5 |- F/ t h e. A
87		nfra1 2845	. . . . 5 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E
88	86, 87	nfan 1875	. . . 4 |- F/ t ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
89	4, 88	nfan 1875	. . 3 |- F/ t ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
90		eqid 2467	. . 3 |- ( t e. T |-> ( ( h ` t ) + sup ( ran F , RR , `' < ) ) ) = ( t e. T |-> ( ( h ` t ) + sup ( ran F , RR , `' < ) ) )
91	44	adantr 465	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> F : T --> RR )
92	39	adantr 465	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> sup ( ran F , RR , `' < ) e. RR )
93	18	3adant1r 1221	. . 3 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
94	26	adantlr 714	. . 3 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ x e. RR ) -> ( t e. T |-> x ) e. A )
95		stoweidlem62.2	. . . . 5 |- F/ f ph
96	10	sseld 3503	. . . . . . . 8 |- ( ph -> ( f e. A -> f e. C ) )
97	9	eleq2i 2545	. . . . . . . 8 |- ( f e. C <-> f e. ( J Cn K ) )
98	96, 97	syl6ib 226	. . . . . . 7 |- ( ph -> ( f e. A -> f e. ( J Cn K ) ) )
99		eqid 2467	. . . . . . . 8 |- U. J = U. J
100		uniretop 21020	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
101	5	unieqi 4254	. . . . . . . . 9 |- U. K = U. ( topGen ` ran (,) )
102	100, 101	eqtr4i 2499	. . . . . . . 8 |- RR = U. K
103	99, 102	cnf 19529	. . . . . . 7 |- ( f e. ( J Cn K ) -> f : U. J --> RR )
104	98, 103	syl6 33	. . . . . 6 |- ( ph -> ( f e. A -> f : U. J --> RR ) )
105		feq2 5713	. . . . . . 7 |- ( T = U. J -> ( f : T --> RR <-> f : U. J --> RR ) )
106	7, 105	mp1i 12	. . . . . 6 |- ( ph -> ( f : T --> RR <-> f : U. J --> RR ) )
107	104, 106	sylibrd 234	. . . . 5 |- ( ph -> ( f e. A -> f : T --> RR ) )
108	95, 107	ralrimi 2864	. . . 4 |- ( ph -> A. f e. A f : T --> RR )
109	108	adantr 465	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. f e. A f : T --> RR )
110		simprl 755	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> h e. A )
111	52	eqcomd 2475	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) = ( H ` t ) )
112	111	oveq2d 6299	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) = ( ( h ` t ) - ( H ` t ) ) )
113	112	fveq2d 5869	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
114	113	adantlr 714	. . . . . 6 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
115		simplrr 760	. . . . . . 7 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
116		rsp 2830	. . . . . . . 8 |- ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E -> ( t e. T -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
117	116	imp 429	. . . . . . 7 |- ( ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
118	115, 117	sylancom 667	. . . . . 6 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
119	114, 118	eqbrtrd 4467	. . . . 5 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) ) < E )
120	119	ex 434	. . . 4 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> ( t e. T -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) ) < E ) )
121	89, 120	ralrimi 2864	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. t e. T ( abs ` ( ( h ` t ) - ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) ) < E )
122	84, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 121	stoweidlem21 31337	. 2 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )
123	83, 122	rexlimddv 2959	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   F/wnf 1599   e. wcel 1767   F/_wnfc 2615   =/= wne 2662   A.wral 2814   E.wrex 2815   C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   `'ccnv 4998   ran crn 5000   -->wf 5583   ` cfv 5587  (class class class)co 6283   supcsup 7899   RRcr 9490   0cc0 9491   1c1 9492   + caddc 9494   x. cmul 9496   < clt 9627   <_ cle 9628   - cmin 9804   -ucneg 9805   / cdiv 10205   2c2 10584   3c3 10585   RR+crp 11219   (,)cioo 11528   abscabs 13029   topGenctg 14692   Topctop 19177   Cn ccn 19507   Compccmp 19668

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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571

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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-cn 19510  df-cnp 19511  df-cmp 19669  df-tx 19814  df-hmeo 20007  df-xms 20574  df-ms 20575  df-tms 20576

This theorem is referenced by:  stoweid  31379