Theorem stoweidlem62 38035

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.) (Revised by AV, 13-Sep-2020.)

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 ) - inf ( 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 ) - inf ( ran F , RR , < ) ) )
2		nfmpt1 4485	. . . . 5 |- F/_ t ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) )
3	1, 2	nfcxfr 2610	. . . 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 2537	. . . . . . 7 |- ( g = h -> ( g e. A <-> h e. A ) )
12	11	3anbi3d 1371	. . . . . 6 |- ( g = h -> ( ( ph /\ f e. A /\ g e. A ) <-> ( ph /\ f e. A /\ h e. A ) ) )
13		fveq1 5878	. . . . . . . . 9 |- ( g = h -> ( g ` t ) = ( h ` t ) )
14	13	oveq2d 6324	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) + ( g ` t ) ) = ( ( f ` t ) + ( h ` t ) ) )
15	14	mpteq2dv 4483	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) )
16	15	eleq1d 2533	. . . . . 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 327	. . . . 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 2120	. . . 4 |- ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A )
20	13	oveq2d 6324	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) x. ( g ` t ) ) = ( ( f ` t ) x. ( h ` t ) ) )
21	20	mpteq2dv 4483	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) )
22	21	eleq1d 2533	. . . . . 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 327	. . . . 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 2120	. . . 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 5083	. . . . . . 7 |- F/_ t ran F
30		nfcv 2612	. . . . . . 7 |- F/_ t RR
31		nfcv 2612	. . . . . . 7 |- F/_ t <
32	29, 30, 31	nfinf 8016	. . . . . 6 |- F/_ tinf ( ran F , RR , < )
33		eqid 2471	. . . . . 6 |- ( T X. { -uinf ( ran F , RR , < ) } ) = ( T X. { -uinf ( ran F , RR , < ) } )
34		cmptop 20487	. . . . . . 7 |- ( J e. Comp -> J e. Top )
35	6, 34	syl 17	. . . . . 6 |- ( ph -> J e. Top )
36		stoweidlem62.14	. . . . . 6 |- ( ph -> F e. C )
37	36, 9	syl6eleq 2559	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
38	28, 4, 7, 5, 6, 37, 8	stoweidlem29 38001	. . . . . . 7 |- ( ph -> (inf ( ran F , RR , < ) e. ran F /\ inf ( ran F , RR , < ) e. RR /\ A. t e. T inf ( ran F , RR , < ) <_ ( F ` t ) ) )
39	38	simp2d 1043	. . . . . 6 |- ( ph -> inf ( ran F , RR , < ) e. RR )
40	28, 32, 4, 7, 33, 5, 35, 9, 36, 39	stoweidlem47 38020	. . . . 5 |- ( ph -> ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) ) e. C )
41	1, 40	syl5eqel 2553	. . . 4 |- ( ph -> H e. C )
42	38	simp3d 1044	. . . . . . . . 9 |- ( ph -> A. t e. T inf ( ran F , RR , < ) <_ ( F ` t ) )
43	42	r19.21bi 2776	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) <_ ( F ` t ) )
44	5, 7, 9, 36	fcnre 37409	. . . . . . . . . 10 |- ( ph -> F : T --> RR )
45	44	fnvinran 37398	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
46	39	adantr 472	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) e. RR )
47	45, 46	subge0d 10224	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) <-> inf ( ran F , RR , < ) <_ ( F ` t ) ) )
48	43, 47	mpbird 240	. . . . . . 7 |- ( ( ph /\ t e. T ) -> 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) )
49		simpr 468	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
50	45, 46	resubcld 10068	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) e. RR )
51	1	fvmpt2 5972	. . . . . . . 8 |- ( ( t e. T /\ ( ( F ` t ) - inf ( ran F , RR , < ) ) e. RR ) -> ( H ` t ) = ( ( F ` t ) - inf ( ran F , RR , < ) ) )
52	49, 50, 51	syl2anc 673	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) = ( ( F ` t ) - inf ( ran F , RR , < ) ) )
53	48, 52	breqtrrd 4422	. . . . . 6 |- ( ( ph /\ t e. T ) -> 0 <_ ( H ` t ) )
54	53	ex 441	. . . . 5 |- ( ph -> ( t e. T -> 0 <_ ( H ` t ) ) )
55	4, 54	ralrimi 2800	. . . 4 |- ( ph -> A. t e. T 0 <_ ( H ` t ) )
56		stoweidlem62.15	. . . . 5 |- ( ph -> E e. RR+ )
57	56	rphalfcld 11376	. . . 4 |- ( ph -> ( E / 2 ) e. RR+ )
58	56	rpred 11364	. . . . . 6 |- ( ph -> E e. RR )
59	58	rehalfcld 10882	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
60		3re 10705	. . . . . . 7 |- 3 e. RR
61		3ne0 10726	. . . . . . 7 |- 3 =/= 0
62	60, 61	rereccli 10394	. . . . . 6 |- ( 1 / 3 ) e. RR
63	62	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. RR )
64		rphalflt 11352	. . . . . 6 |- ( E e. RR+ -> ( E / 2 ) < E )
65	56, 64	syl 17	. . . . 5 |- ( ph -> ( E / 2 ) < E )
66		stoweidlem62.17	. . . . 5 |- ( ph -> E < ( 1 / 3 ) )
67	59, 58, 63, 65, 66	lttrd 9813	. . . 4 |- ( ph -> ( E / 2 ) < ( 1 / 3 ) )
68	3, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67	stoweidlem61 38034	. . 3 |- ( ph -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
69		nfra1 2785	. . . . . . 7 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) )
70	4, 69	nfan 2031	. . . . . 6 |- F/ t ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
71		rsp 2773	. . . . . . 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 11366	. . . . . . . . . 10 |- ( ph -> E e. CC )
73		2cnd 10704	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
74		2ne0 10724	. . . . . . . . . . 11 |- 2 =/= 0
75	74	a1i 11	. . . . . . . . . 10 |- ( ph -> 2 =/= 0 )
76	72, 73, 75	divcan2d 10407	. . . . . . . . 9 |- ( ph -> ( 2 x. ( E / 2 ) ) = E )
77	76	breq2d 4407	. . . . . . . 8 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) <-> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
78	77	biimpd 212	. . . . . . 7 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
79	71, 78	sylan9r 670	. . . . . 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 2800	. . . . 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 441	. . . 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 2857	. . 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 4485	. . 3 |- F/_ t ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
85		nfcv 2612	. . 3 |- F/_ t h
86		nfv 1769	. . . . 5 |- F/ t h e. A
87		nfra1 2785	. . . . 5 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E
88	86, 87	nfan 2031	. . . 4 |- F/ t ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
89	4, 88	nfan 2031	. . 3 |- F/ t ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
90		eqid 2471	. . 3 |- ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) ) = ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
91	44	adantr 472	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> F : T --> RR )
92	39	adantr 472	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> inf ( ran F , RR , < ) e. RR )
93	18	3adant1r 1285	. . 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 729	. . 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 3417	. . . . . . . 8 |- ( ph -> ( f e. A -> f e. C ) )
97	9	eleq2i 2541	. . . . . . . 8 |- ( f e. C <-> f e. ( J Cn K ) )
98	96, 97	syl6ib 234	. . . . . . 7 |- ( ph -> ( f e. A -> f e. ( J Cn K ) ) )
99		eqid 2471	. . . . . . . 8 |- U. J = U. J
100		uniretop 21861	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
101	5	unieqi 4199	. . . . . . . . 9 |- U. K = U. ( topGen ` ran (,) )
102	100, 101	eqtr4i 2496	. . . . . . . 8 |- RR = U. K
103	99, 102	cnf 20339	. . . . . . 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 5721	. . . . . . 7 |- ( T = U. J -> ( f : T --> RR <-> f : U. J --> RR ) )
106	7, 105	mp1i 13	. . . . . 6 |- ( ph -> ( f : T --> RR <-> f : U. J --> RR ) )
107	104, 106	sylibrd 242	. . . . 5 |- ( ph -> ( f e. A -> f : T --> RR ) )
108	95, 107	ralrimi 2800	. . . 4 |- ( ph -> A. f e. A f : T --> RR )
109	108	adantr 472	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. f e. A f : T --> RR )
110		simprl 772	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> h e. A )
111	52	eqcomd 2477	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) = ( H ` t ) )
112	111	oveq2d 6324	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) = ( ( h ` t ) - ( H ` t ) ) )
113	112	fveq2d 5883	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
114	113	adantlr 729	. . . . . 6 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
115		simplrr 779	. . . . . . 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		rspa 2774	. . . . . . 7 |- ( ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
117	115, 116	sylancom 680	. . . . . 6 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
118	114, 117	eqbrtrd 4416	. . . . 5 |- ( ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) < E )
119	118	ex 441	. . . 4 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> ( t e. T -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) < E ) )
120	89, 119	ralrimi 2800	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. t e. T ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) < E )
121	84, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 120	stoweidlem21 37993	. 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 )
122	83, 121	rexlimddv 2875	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   F/wnf 1675   e. wcel 1904   F/_wnfc 2599   =/= wne 2641   A.wral 2756   E.wrex 2757   C_ wss 3390   (/)c0 3722   {csn 3959   U.cuni 4190   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308  infcinf 7973   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   -ucneg 9881   / cdiv 10291   2c2 10681   3c3 10682   RR+crp 11325   (,)cioo 11660   abscabs 13374   topGenctg 15414   Topctop 19994   Cn ccn 20317   Compccmp 20478

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-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

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-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-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-ioc 11665  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-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415

This theorem is referenced by:  stoweid  38037