Theorem stoweidlem62 40279

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 4747	. . . . 5 |- F/_ t ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) )
3	1, 2	nfcxfr 2762	. . . 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 2689	. . . . . . 7 |- ( g = h -> ( g e. A <-> h e. A ) )
12	11	3anbi3d 1405	. . . . . 6 |- ( g = h -> ( ( ph /\ f e. A /\ g e. A ) <-> ( ph /\ f e. A /\ h e. A ) ) )
13		fveq1 6190	. . . . . . . . 9 |- ( g = h -> ( g ` t ) = ( h ` t ) )
14	13	oveq2d 6666	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) + ( g ` t ) ) = ( ( f ` t ) + ( h ` t ) ) )
15	14	mpteq2dv 4745	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) )
16	15	eleq1d 2686	. . . . . 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 334	. . . . 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 2263	. . . 4 |- ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A )
20	13	oveq2d 6666	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) x. ( g ` t ) ) = ( ( f ` t ) x. ( h ` t ) ) )
21	20	mpteq2dv 4745	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) )
22	21	eleq1d 2686	. . . . . 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 334	. . . . 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 2263	. . . 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 5368	. . . . . . 7 |- F/_ t ran F
30		nfcv 2764	. . . . . . 7 |- F/_ t RR
31		nfcv 2764	. . . . . . 7 |- F/_ t <
32	29, 30, 31	nfinf 8388	. . . . . 6 |- F/_ tinf ( ran F , RR , < )
33		eqid 2622	. . . . . 6 |- ( T X. { -uinf ( ran F , RR , < ) } ) = ( T X. { -uinf ( ran F , RR , < ) } )
34		cmptop 21198	. . . . . . 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 2711	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
38	28, 4, 7, 5, 6, 37, 8	stoweidlem29 40246	. . . . . . 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 1074	. . . . . 6 |- ( ph -> inf ( ran F , RR , < ) e. RR )
40	28, 32, 4, 7, 33, 5, 35, 9, 36, 39	stoweidlem47 40264	. . . . 5 |- ( ph -> ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) ) e. C )
41	1, 40	syl5eqel 2705	. . . 4 |- ( ph -> H e. C )
42	38	simp3d 1075	. . . . . . . . 9 |- ( ph -> A. t e. T inf ( ran F , RR , < ) <_ ( F ` t ) )
43	42	r19.21bi 2932	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) <_ ( F ` t ) )
44	5, 7, 9, 36	fcnre 39184	. . . . . . . . . 10 |- ( ph -> F : T --> RR )
45	44	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
46	39	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) e. RR )
47	45, 46	subge0d 10617	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) <-> inf ( ran F , RR , < ) <_ ( F ` t ) ) )
48	43, 47	mpbird 247	. . . . . . 7 |- ( ( ph /\ t e. T ) -> 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) )
49		simpr 477	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
50	45, 46	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) e. RR )
51	1	fvmpt2 6291	. . . . . . . 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 693	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) = ( ( F ` t ) - inf ( ran F , RR , < ) ) )
53	48, 52	breqtrrd 4681	. . . . . 6 |- ( ( ph /\ t e. T ) -> 0 <_ ( H ` t ) )
54	53	ex 450	. . . . 5 |- ( ph -> ( t e. T -> 0 <_ ( H ` t ) ) )
55	4, 54	ralrimi 2957	. . . 4 |- ( ph -> A. t e. T 0 <_ ( H ` t ) )
56		stoweidlem62.15	. . . . 5 |- ( ph -> E e. RR+ )
57	56	rphalfcld 11884	. . . 4 |- ( ph -> ( E / 2 ) e. RR+ )
58	56	rpred 11872	. . . . . 6 |- ( ph -> E e. RR )
59	58	rehalfcld 11279	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
60		3re 11094	. . . . . . 7 |- 3 e. RR
61		3ne0 11115	. . . . . . 7 |- 3 =/= 0
62	60, 61	rereccli 10790	. . . . . 6 |- ( 1 / 3 ) e. RR
63	62	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. RR )
64		rphalflt 11860	. . . . . 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 10198	. . . 4 |- ( ph -> ( E / 2 ) < ( 1 / 3 ) )
68	3, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67	stoweidlem61 40278	. . 3 |- ( ph -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
69		nfra1 2941	. . . . . . 7 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) )
70	4, 69	nfan 1828	. . . . . 6 |- F/ t ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
71		rsp 2929	. . . . . . 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 11874	. . . . . . . . . 10 |- ( ph -> E e. CC )
73		2cnd 11093	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
74		2ne0 11113	. . . . . . . . . . 11 |- 2 =/= 0
75	74	a1i 11	. . . . . . . . . 10 |- ( ph -> 2 =/= 0 )
76	72, 73, 75	divcan2d 10803	. . . . . . . . 9 |- ( ph -> ( 2 x. ( E / 2 ) ) = E )
77	76	breq2d 4665	. . . . . . . 8 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) <-> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
78	77	biimpd 219	. . . . . . 7 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
79	71, 78	sylan9r 690	. . . . . 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 2957	. . . . 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 450	. . . 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 3016	. . 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 4747	. . 3 |- F/_ t ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
85		nfcv 2764	. . 3 |- F/_ t h
86		nfv 1843	. . . . 5 |- F/ t h e. A
87		nfra1 2941	. . . . 5 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E
88	86, 87	nfan 1828	. . . 4 |- F/ t ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
89	4, 88	nfan 1828	. . 3 |- F/ t ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
90		eqid 2622	. . 3 |- ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) ) = ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
91	44	adantr 481	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> F : T --> RR )
92	39	adantr 481	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> inf ( ran F , RR , < ) e. RR )
93	18	3adant1r 1319	. . 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 751	. . 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 3602	. . . . . . . 8 |- ( ph -> ( f e. A -> f e. C ) )
97	9	eleq2i 2693	. . . . . . . 8 |- ( f e. C <-> f e. ( J Cn K ) )
98	96, 97	syl6ib 241	. . . . . . 7 |- ( ph -> ( f e. A -> f e. ( J Cn K ) ) )
99		eqid 2622	. . . . . . . 8 |- U. J = U. J
100		uniretop 22566	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
101	5	unieqi 4445	. . . . . . . . 9 |- U. K = U. ( topGen ` ran (,) )
102	100, 101	eqtr4i 2647	. . . . . . . 8 |- RR = U. K
103	99, 102	cnf 21050	. . . . . . 7 |- ( f e. ( J Cn K ) -> f : U. J --> RR )
104	98, 103	syl6 35	. . . . . 6 |- ( ph -> ( f e. A -> f : U. J --> RR ) )
105		feq2 6027	. . . . . . 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 249	. . . . 5 |- ( ph -> ( f e. A -> f : T --> RR ) )
108	95, 107	ralrimi 2957	. . . 4 |- ( ph -> A. f e. A f : T --> RR )
109	108	adantr 481	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. f e. A f : T --> RR )
110		simprl 794	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> h e. A )
111	52	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) = ( H ` t ) )
112	111	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) = ( ( h ` t ) - ( H ` t ) ) )
113	112	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
114	113	adantlr 751	. . . . . 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 801	. . . . . . 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 2930	. . . . . . 7 |- ( ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
117	115, 116	sylancom 701	. . . . . 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 4675	. . . . 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 450	. . . 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 2957	. . 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 40238	. 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 3035	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   3c3 11071   RR+crp 11832   (,)cioo 12175   abscabs 13974   topGenctg 16098   Topctop 20698   Cn ccn 21028   Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  stoweid  40280