Theorem dyadmbllem 23367

Description: Lemma for dyadmbl 23368. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypotheses

Ref	Expression
dyadmbl.1	|- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )
dyadmbl.2	|- G = { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) }
dyadmbl.3	|- ( ph -> A C_ ran F )

Assertion

Ref	Expression
dyadmbllem	|- ( ph -> U. ( [,] " A ) = U. ( [,] " G ) )

Distinct variable groups:   x, y   z, w, ph   x, w, y, A, z   z, G   w, F, x, y, z

Allowed substitution hints:   ph( x, y)   G( x, y, w)

Proof of Theorem dyadmbllem

Dummy variables a m t i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4440	. . . 4 |- ( a e. U. ( [,] " A ) <-> E. i e. ( [,] " A ) a e. i )
2		iccf 12272	. . . . . . 7 |- [,] : ( RR* X. RR* ) --> ~P RR*
3		ffn 6045	. . . . . . 7 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 5	. . . . . 6 |- [,] Fn ( RR* X. RR* )
5		dyadmbl.3	. . . . . . 7 |- ( ph -> A C_ ran F )
6		dyadmbl.1	. . . . . . . . . 10 |- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )
7	6	dyadf 23359	. . . . . . . . 9 |- F : ( ZZ X. NN0 ) --> ( <_ i^i ( RR X. RR ) )
8		frn 6053	. . . . . . . . 9 |- ( F : ( ZZ X. NN0 ) --> ( <_ i^i ( RR X. RR ) ) -> ran F C_ ( <_ i^i ( RR X. RR ) ) )
9	7, 8	ax-mp 5	. . . . . . . 8 |- ran F C_ ( <_ i^i ( RR X. RR ) )
10		inss2 3834	. . . . . . . . 9 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
11		rexpssxrxp 10084	. . . . . . . . 9 |- ( RR X. RR ) C_ ( RR* X. RR* )
12	10, 11	sstri 3612	. . . . . . . 8 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR* X. RR* )
13	9, 12	sstri 3612	. . . . . . 7 |- ran F C_ ( RR* X. RR* )
14	5, 13	syl6ss 3615	. . . . . 6 |- ( ph -> A C_ ( RR* X. RR* ) )
15		eleq2 2690	. . . . . . 7 |- ( i = ( [,] ` t ) -> ( a e. i <-> a e. ( [,] ` t ) ) )
16	15	rexima 6497	. . . . . 6 |- ( ( [,] Fn ( RR* X. RR* ) /\ A C_ ( RR* X. RR* ) ) -> ( E. i e. ( [,] " A ) a e. i <-> E. t e. A a e. ( [,] ` t ) ) )
17	4, 14, 16	sylancr 695	. . . . 5 |- ( ph -> ( E. i e. ( [,] " A ) a e. i <-> E. t e. A a e. ( [,] ` t ) ) )
18		ssrab2 3687	. . . . . . . . 9 |- { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } C_ A
19	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> A C_ ran F )
20	18, 19	syl5ss 3614	. . . . . . . 8 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } C_ ran F )
21		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> t e. A )
22		ssid 3624	. . . . . . . . . 10 |- ( [,] ` t ) C_ ( [,] ` t )
23		fveq2 6191	. . . . . . . . . . . 12 |- ( a = t -> ( [,] ` a ) = ( [,] ` t ) )
24	23	sseq2d 3633	. . . . . . . . . . 11 |- ( a = t -> ( ( [,] ` t ) C_ ( [,] ` a ) <-> ( [,] ` t ) C_ ( [,] ` t ) ) )
25	24	rspcev 3309	. . . . . . . . . 10 |- ( ( t e. A /\ ( [,] ` t ) C_ ( [,] ` t ) ) -> E. a e. A ( [,] ` t ) C_ ( [,] ` a ) )
26	21, 22, 25	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> E. a e. A ( [,] ` t ) C_ ( [,] ` a ) )
27		rabn0 3958	. . . . . . . . 9 |- ( { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } =/= (/) <-> E. a e. A ( [,] ` t ) C_ ( [,] ` a ) )
28	26, 27	sylibr 224	. . . . . . . 8 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } =/= (/) )
29	6	dyadmax 23366	. . . . . . . 8 |- ( ( { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } C_ ran F /\ { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } =/= (/) ) -> E. m e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) )
30	20, 28, 29	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> E. m e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) )
31		fveq2 6191	. . . . . . . . . . 11 |- ( a = m -> ( [,] ` a ) = ( [,] ` m ) )
32	31	sseq2d 3633	. . . . . . . . . 10 |- ( a = m -> ( ( [,] ` t ) C_ ( [,] ` a ) <-> ( [,] ` t ) C_ ( [,] ` m ) ) )
33	32	elrab 3363	. . . . . . . . 9 |- ( m e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } <-> ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) )
34		simprlr 803	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> ( [,] ` t ) C_ ( [,] ` m ) )
35		simplrr 801	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> a e. ( [,] ` t ) )
36	34, 35	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> a e. ( [,] ` m ) )
37		simprll 802	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> m e. A )
38		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( a = w -> ( [,] ` a ) = ( [,] ` w ) )
39	38	sseq2d 3633	. . . . . . . . . . . . . . . . . . 19 |- ( a = w -> ( ( [,] ` t ) C_ ( [,] ` a ) <-> ( [,] ` t ) C_ ( [,] ` w ) ) )
40	39	elrab 3363	. . . . . . . . . . . . . . . . . 18 |- ( w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } <-> ( w e. A /\ ( [,] ` t ) C_ ( [,] ` w ) ) )
41	40	imbi1i 339	. . . . . . . . . . . . . . . . 17 |- ( ( w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) <-> ( ( w e. A /\ ( [,] ` t ) C_ ( [,] ` w ) ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
42		impexp 462	. . . . . . . . . . . . . . . . 17 |- ( ( ( w e. A /\ ( [,] ` t ) C_ ( [,] ` w ) ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) <-> ( w e. A -> ( ( [,] ` t ) C_ ( [,] ` w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) )
43	41, 42	bitri 264	. . . . . . . . . . . . . . . 16 |- ( ( w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) <-> ( w e. A -> ( ( [,] ` t ) C_ ( [,] ` w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) )
44		impexp 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( [,] ` t ) C_ ( [,] ` w ) /\ ( [,] ` m ) C_ ( [,] ` w ) ) -> m = w ) <-> ( ( [,] ` t ) C_ ( [,] ` w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
45		sstr2 3610	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( [,] ` t ) C_ ( [,] ` m ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> ( [,] ` t ) C_ ( [,] ` w ) ) )
46	45	ad2antll 765	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> ( [,] ` t ) C_ ( [,] ` w ) ) )
47	46	ancrd 577	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> ( ( [,] ` t ) C_ ( [,] ` w ) /\ ( [,] ` m ) C_ ( [,] ` w ) ) ) )
48	47	imim1d 82	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( ( ( [,] ` t ) C_ ( [,] ` w ) /\ ( [,] ` m ) C_ ( [,] ` w ) ) -> m = w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
49	44, 48	syl5bir 233	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( ( [,] ` t ) C_ ( [,] ` w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
50	49	imim2d 57	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( w e. A -> ( ( [,] ` t ) C_ ( [,] ` w ) -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> ( w e. A -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) )
51	43, 50	syl5bi 232	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( ( w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) -> ( w e. A -> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) )
52	51	ralimdv2 2961	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) ) -> ( A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) -> A. w e. A ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
53	52	impr 649	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> A. w e. A ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) )
54		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( z = m -> ( [,] ` z ) = ( [,] ` m ) )
55	54	sseq1d 3632	. . . . . . . . . . . . . . . 16 |- ( z = m -> ( ( [,] ` z ) C_ ( [,] ` w ) <-> ( [,] ` m ) C_ ( [,] ` w ) ) )
56		equequ1 1952	. . . . . . . . . . . . . . . 16 |- ( z = m -> ( z = w <-> m = w ) )
57	55, 56	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( z = m -> ( ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
58	57	ralbidv 2986	. . . . . . . . . . . . . 14 |- ( z = m -> ( A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) <-> A. w e. A ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
59		dyadmbl.2	. . . . . . . . . . . . . 14 |- G = { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) }
60	58, 59	elrab2 3366	. . . . . . . . . . . . 13 |- ( m e. G <-> ( m e. A /\ A. w e. A ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) )
61	37, 53, 60	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> m e. G )
62		ffun 6048	. . . . . . . . . . . . . 14 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> Fun [,] )
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 |- Fun [,]
64		ssrab2 3687	. . . . . . . . . . . . . . . . 17 |- { z e. A | A. w e. A ( ( [,] ` z ) C_ ( [,] ` w ) -> z = w ) } C_ A
65	59, 64	eqsstri 3635	. . . . . . . . . . . . . . . 16 |- G C_ A
66	65, 14	syl5ss 3614	. . . . . . . . . . . . . . 15 |- ( ph -> G C_ ( RR* X. RR* ) )
67	2	fdmi 6052	. . . . . . . . . . . . . . 15 |- dom [,] = ( RR* X. RR* )
68	66, 67	syl6sseqr 3652	. . . . . . . . . . . . . 14 |- ( ph -> G C_ dom [,] )
69	68	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> G C_ dom [,] )
70		funfvima2 6493	. . . . . . . . . . . . 13 |- ( ( Fun [,] /\ G C_ dom [,] ) -> ( m e. G -> ( [,] ` m ) e. ( [,] " G ) ) )
71	63, 69, 70	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> ( m e. G -> ( [,] ` m ) e. ( [,] " G ) ) )
72	61, 71	mpd 15	. . . . . . . . . . 11 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> ( [,] ` m ) e. ( [,] " G ) )
73		elunii 4441	. . . . . . . . . . 11 |- ( ( a e. ( [,] ` m ) /\ ( [,] ` m ) e. ( [,] " G ) ) -> a e. U. ( [,] " G ) )
74	36, 72, 73	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) /\ ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) /\ A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) ) ) -> a e. U. ( [,] " G ) )
75	74	exp32 631	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> ( ( m e. A /\ ( [,] ` t ) C_ ( [,] ` m ) ) -> ( A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) -> a e. U. ( [,] " G ) ) ) )
76	33, 75	syl5bi 232	. . . . . . . 8 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> ( m e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } -> ( A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) -> a e. U. ( [,] " G ) ) ) )
77	76	rexlimdv 3030	. . . . . . 7 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> ( E. m e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } A. w e. { a e. A | ( [,] ` t ) C_ ( [,] ` a ) } ( ( [,] ` m ) C_ ( [,] ` w ) -> m = w ) -> a e. U. ( [,] " G ) ) )
78	30, 77	mpd 15	. . . . . 6 |- ( ( ph /\ ( t e. A /\ a e. ( [,] ` t ) ) ) -> a e. U. ( [,] " G ) )
79	78	rexlimdvaa 3032	. . . . 5 |- ( ph -> ( E. t e. A a e. ( [,] ` t ) -> a e. U. ( [,] " G ) ) )
80	17, 79	sylbid 230	. . . 4 |- ( ph -> ( E. i e. ( [,] " A ) a e. i -> a e. U. ( [,] " G ) ) )
81	1, 80	syl5bi 232	. . 3 |- ( ph -> ( a e. U. ( [,] " A ) -> a e. U. ( [,] " G ) ) )
82	81	ssrdv 3609	. 2 |- ( ph -> U. ( [,] " A ) C_ U. ( [,] " G ) )
83		imass2 5501	. . . 4 |- ( G C_ A -> ( [,] " G ) C_ ( [,] " A ) )
84	65, 83	ax-mp 5	. . 3 |- ( [,] " G ) C_ ( [,] " A )
85		uniss 4458	. . 3 |- ( ( [,] " G ) C_ ( [,] " A ) -> U. ( [,] " G ) C_ U. ( [,] " A ) )
86	84, 85	mp1i 13	. 2 |- ( ph -> U. ( [,] " G ) C_ U. ( [,] " A ) )
87	82, 86	eqssd 3620	1 |- ( ph -> U. ( [,] " A ) = U. ( [,] " G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   <.cop 4183   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   <_ cle 10075   / cdiv 10684   2c2 11070   NN0cn0 11292   ZZcz 11377   [,]cicc 12178   ^cexp 12860

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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233

This theorem is referenced by:  dyadmbl  23368  mblfinlem1  33446  mblfinlem2  33447