Theorem stoweidlem62 37917

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 4491	. . . . 5 |- F/_ t ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) )
3	1, 2	nfcxfr 2589	. . . 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 2516	. . . . . . 7 |- ( g = h -> ( g e. A <-> h e. A ) )
12	11	3anbi3d 1344	. . . . . 6 |- ( g = h -> ( ( ph /\ f e. A /\ g e. A ) <-> ( ph /\ f e. A /\ h e. A ) ) )
13		fveq1 5862	. . . . . . . . 9 |- ( g = h -> ( g ` t ) = ( h ` t ) )
14	13	oveq2d 6304	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) + ( g ` t ) ) = ( ( f ` t ) + ( h ` t ) ) )
15	14	mpteq2dv 4489	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) )
16	15	eleq1d 2512	. . . . . 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 322	. . . . 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 2106	. . . 4 |- ( ( ph /\ f e. A /\ h e. A ) -> ( t e. T |-> ( ( f ` t ) + ( h ` t ) ) ) e. A )
20	13	oveq2d 6304	. . . . . . . 8 |- ( g = h -> ( ( f ` t ) x. ( g ` t ) ) = ( ( f ` t ) x. ( h ` t ) ) )
21	20	mpteq2dv 4489	. . . . . . 7 |- ( g = h -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) = ( t e. T |-> ( ( f ` t ) x. ( h ` t ) ) ) )
22	21	eleq1d 2512	. . . . . 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 322	. . . . 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 2106	. . . 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 5076	. . . . . . 7 |- F/_ t ran F
30		nfcv 2591	. . . . . . 7 |- F/_ t RR
31		nfcv 2591	. . . . . . 7 |- F/_ t <
32	29, 30, 31	nfinf 7995	. . . . . 6 |- F/_ tinf ( ran F , RR , < )
33		eqid 2450	. . . . . 6 |- ( T X. { -uinf ( ran F , RR , < ) } ) = ( T X. { -uinf ( ran F , RR , < ) } )
34		cmptop 20403	. . . . . . 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 2538	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
38	28, 4, 7, 5, 6, 37, 8	stoweidlem29 37883	. . . . . . 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 1020	. . . . . 6 |- ( ph -> inf ( ran F , RR , < ) e. RR )
40	28, 32, 4, 7, 33, 5, 35, 9, 36, 39	stoweidlem47 37902	. . . . 5 |- ( ph -> ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) ) e. C )
41	1, 40	syl5eqel 2532	. . . 4 |- ( ph -> H e. C )
42	38	simp3d 1021	. . . . . . . . 9 |- ( ph -> A. t e. T inf ( ran F , RR , < ) <_ ( F ` t ) )
43	42	r19.21bi 2756	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) <_ ( F ` t ) )
44	5, 7, 9, 36	fcnre 37340	. . . . . . . . . 10 |- ( ph -> F : T --> RR )
45	44	fnvinran 37329	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
46	39	adantr 467	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> inf ( ran F , RR , < ) e. RR )
47	45, 46	subge0d 10200	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) <-> inf ( ran F , RR , < ) <_ ( F ` t ) ) )
48	43, 47	mpbird 236	. . . . . . 7 |- ( ( ph /\ t e. T ) -> 0 <_ ( ( F ` t ) - inf ( ran F , RR , < ) ) )
49		simpr 463	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
50	45, 46	resubcld 10044	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) e. RR )
51	1	fvmpt2 5955	. . . . . . . 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 666	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) = ( ( F ` t ) - inf ( ran F , RR , < ) ) )
53	48, 52	breqtrrd 4428	. . . . . 6 |- ( ( ph /\ t e. T ) -> 0 <_ ( H ` t ) )
54	53	ex 436	. . . . 5 |- ( ph -> ( t e. T -> 0 <_ ( H ` t ) ) )
55	4, 54	ralrimi 2787	. . . 4 |- ( ph -> A. t e. T 0 <_ ( H ` t ) )
56		stoweidlem62.15	. . . . 5 |- ( ph -> E e. RR+ )
57	56	rphalfcld 11350	. . . 4 |- ( ph -> ( E / 2 ) e. RR+ )
58	56	rpred 11338	. . . . . 6 |- ( ph -> E e. RR )
59	58	rehalfcld 10856	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
60		3re 10680	. . . . . . 7 |- 3 e. RR
61		3ne0 10701	. . . . . . 7 |- 3 =/= 0
62	60, 61	rereccli 10369	. . . . . 6 |- ( 1 / 3 ) e. RR
63	62	a1i 11	. . . . 5 |- ( ph -> ( 1 / 3 ) e. RR )
64		rphalflt 11326	. . . . . 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 9793	. . . 4 |- ( ph -> ( E / 2 ) < ( 1 / 3 ) )
68	3, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67	stoweidlem61 37916	. . 3 |- ( ph -> E. h e. A A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
69		nfra1 2768	. . . . . . 7 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) )
70	4, 69	nfan 2010	. . . . . 6 |- F/ t ( ph /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) )
71		rsp 2753	. . . . . . 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 11340	. . . . . . . . . 10 |- ( ph -> E e. CC )
73		2cnd 10679	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
74		2ne0 10699	. . . . . . . . . . 11 |- 2 =/= 0
75	74	a1i 11	. . . . . . . . . 10 |- ( ph -> 2 =/= 0 )
76	72, 73, 75	divcan2d 10382	. . . . . . . . 9 |- ( ph -> ( 2 x. ( E / 2 ) ) = E )
77	76	breq2d 4413	. . . . . . . 8 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) <-> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
78	77	biimpd 211	. . . . . . 7 |- ( ph -> ( ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < ( 2 x. ( E / 2 ) ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
79	71, 78	sylan9r 663	. . . . . 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 2787	. . . . 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 436	. . . 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 2860	. . 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 4491	. . 3 |- F/_ t ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
85		nfcv 2591	. . 3 |- F/_ t h
86		nfv 1760	. . . . 5 |- F/ t h e. A
87		nfra1 2768	. . . . 5 |- F/ t A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E
88	86, 87	nfan 2010	. . . 4 |- F/ t ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
89	4, 88	nfan 2010	. . 3 |- F/ t ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) )
90		eqid 2450	. . 3 |- ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) ) = ( t e. T |-> ( ( h ` t ) + inf ( ran F , RR , < ) ) )
91	44	adantr 467	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> F : T --> RR )
92	39	adantr 467	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> inf ( ran F , RR , < ) e. RR )
93	18	3adant1r 1260	. . 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 720	. . 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 3430	. . . . . . . 8 |- ( ph -> ( f e. A -> f e. C ) )
97	9	eleq2i 2520	. . . . . . . 8 |- ( f e. C <-> f e. ( J Cn K ) )
98	96, 97	syl6ib 230	. . . . . . 7 |- ( ph -> ( f e. A -> f e. ( J Cn K ) ) )
99		eqid 2450	. . . . . . . 8 |- U. J = U. J
100		uniretop 21776	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
101	5	unieqi 4206	. . . . . . . . 9 |- U. K = U. ( topGen ` ran (,) )
102	100, 101	eqtr4i 2475	. . . . . . . 8 |- RR = U. K
103	99, 102	cnf 20255	. . . . . . 7 |- ( f e. ( J Cn K ) -> f : U. J --> RR )
104	98, 103	syl6 34	. . . . . 6 |- ( ph -> ( f e. A -> f : U. J --> RR ) )
105		feq2 5709	. . . . . . 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 238	. . . . 5 |- ( ph -> ( f e. A -> f : T --> RR ) )
108	95, 107	ralrimi 2787	. . . 4 |- ( ph -> A. f e. A f : T --> RR )
109	108	adantr 467	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> A. f e. A f : T --> RR )
110		simprl 763	. . 3 |- ( ( ph /\ ( h e. A /\ A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E ) ) -> h e. A )
111	52	eqcomd 2456	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) - inf ( ran F , RR , < ) ) = ( H ` t ) )
112	111	oveq2d 6304	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) = ( ( h ` t ) - ( H ` t ) ) )
113	112	fveq2d 5867	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( ( F ` t ) - inf ( ran F , RR , < ) ) ) ) = ( abs ` ( ( h ` t ) - ( H ` t ) ) ) )
114	113	adantlr 720	. . . . . 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 770	. . . . . . 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 2754	. . . . . . 7 |- ( ( A. t e. T ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E /\ t e. T ) -> ( abs ` ( ( h ` t ) - ( H ` t ) ) ) < E )
117	115, 116	sylancom 672	. . . . . 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 4422	. . . . 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 436	. . . 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 2787	. . 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 37875	. 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 2882	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   F/wnf 1666   e. wcel 1886   F/_wnfc 2578   =/= wne 2621   A.wral 2736   E.wrex 2737   C_ wss 3403   (/)c0 3730   {csn 3967   U.cuni 4197   class class class wbr 4401   |-> cmpt 4460   X. cxp 4831   ran crn 4834   -->wf 5577   ` cfv 5581  (class class class)co 6288  infcinf 7952   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   < clt 9672   <_ cle 9673   - cmin 9857   -ucneg 9858   / cdiv 10266   2c2 10656   3c3 10657   RR+crp 11299   (,)cioo 11632   abscabs 13290   topGenctg 15329   Topctop 19910   Cn ccn 20233   Compccmp 20394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330

This theorem is referenced by:  stoweid  37919