Theorem bcthlem30 8028

Description: Lemma for bcth 8032. Apply the Axiom of Dependent Choice acdc5 7493 to show the existence of the recursive sequence of balls g.

Hypotheses

Ref	Expression
bcthlem29.1	|- D e. CMet
bcthlem29.3	|- X = dom dom D
bcthlem29.4	|- J = (Open` D)
bcthlem29.5	|- M:NN-->P~X
bcthlem29.6	|- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))})}
bcthlem29.7	|- A = (X X. {x e. RR | 0 < x})
bcthlem29.8	|- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))

Assertion

Ref	Expression
bcthlem30	|- ((A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ Q e. A) -> E.g(g:NN-->A /\ (g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))

Distinct variable groups:   g,j,k,p,y,z,A   g,r,x,D,j,k,p,y,z   g,F,k   g,J,j,p,r,x,y,z   g,M,j,p,r,x,y,z   O,p,r,z   Q,g   g,X,j,k,p,r,x,y,z

Proof of Theorem bcthlem30

Step	Hyp	Ref	Expression
1		bcthlem29.7	. . . 4 |- A = (X X. {x e. RR | 0 < x})
2		bcthlem29.3	. . . . . 6 |- X = dom dom D
3		bcthlem29.1	. . . . . . . 8 |- D e. CMet
4		dmexg 3358	. . . . . . . 8 |- (D e. CMet -> dom D e. V)
5	3, 4	ax-mp 7	. . . . . . 7 |- dom D e. V
6	5	dmex 3360	. . . . . 6 |- dom dom D e. V
7	2, 6	eqeltr 1544	. . . . 5 |- X e. V
8		reex 5312	. . . . . 6 |- RR e. V
9	8	rabex 2725	. . . . 5 |- {x e. RR | 0 < x} e. V
10	7, 9	xpex 3260	. . . 4 |- (X X. {x e. RR | 0 < x}) e. V
11	1, 10	eqeltr 1544	. . 3 |- A e. V
12	11	acdc5 7493	. 2 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ Q e. A) -> E.g(g:NN-->A /\ (g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))
13		bcthlem29.4	. . . . . . . . 9 |- J = (Open` D)
14		bcthlem29.8	. . . . . . . . 9 |- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))
15	3, 2, 13, 1, 14	bcthlem14 8012	. . . . . . . 8 |- ((((M` j) (_ X /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ (j e. NN /\ y e. A)) -> {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} =/= (/))
16		bcthlem29.5	. . . . . . . . . . . 12 |- M:NN-->P~X
17	16	ffvelrni 3815	. . . . . . . . . . 11 |- (j e. NN -> (M` j) e. P~X)
18		elpwi 2406	. . . . . . . . . . 11 |- ((M` j) e. P~X -> (M` j) (_ X)
19	17, 18	syl 10	. . . . . . . . . 10 |- (j e. NN -> (M` j) (_ X)
20	19	anim1i 334	. . . . . . . . 9 |- ((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) -> ((M` j) (_ X /\ ((int` J)` ((cls` J)` (M` j))) = (/)))
21	20	adantr 389	. . . . . . . 8 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> ((M` j) (_ X /\ ((int` J)` ((cls` J)` (M` j))) = (/)))
22		simpll 412	. . . . . . . . 9 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> j e. NN)
23		pm3.27 323	. . . . . . . . 9 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> y e. A)
24	22, 23	jca 288	. . . . . . . 8 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> (j e. NN /\ y e. A))
25	15, 21, 24	sylanc 471	. . . . . . 7 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} =/= (/))
26	3, 2, 1	bcthlem12 8010	. . . . . . 7 |- {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. P~A
27	25, 26	jctil 292	. . . . . 6 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> ({<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. P~A /\ {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} =/= (/)))
28		eldifsn 2462	. . . . . 6 |- ({<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. (P~A \ {(/)}) <-> ({<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. P~A /\ {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} =/= (/)))
29	27, 28	sylibr 200	. . . . 5 |- (((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) /\ y e. A) -> {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. (P~A \ {(/)}))
30	29	r19.21aiva 1714	. . . 4 |- ((j e. NN /\ ((int` J)` ((cls` J)` (M` j))) = (/)) -> A.y e. A {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. (P~A \ {(/)}))
31	30	r19.20ia 1705	. . 3 |- (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) -> A.j e. NN A.y e. A {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. (P~A \ {(/)}))
32		bcthlem29.6	. . . 4 |- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))})}
33	32	foprab2 4119	. . 3 |- (A.j e. NN A.y e. A {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ O))} e. (P~A \ {(/)}) <-> F:(NN X. A)-->(P~A \ {(/)}))
34	31, 33	sylib 198	. 2 |- (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) -> F:(NN X. A)-->(P~A \ {(/)}))
35	12, 34	sylan 448	1 |- ((A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ Q e. A) -> E.g(g:NN-->A /\ (g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  {crab 1648  Vcvv 1811   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280  P~cpw 2401  {csn 2409   class class class wbr 2619  {copab 2666   X. cxp 3168  dom cdm 3170  -->wf 3178  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  RRcr 5233  0cc0 5234  1c1 5235   + caddc 5237   / cdiv 5294  NNcn 5296   < clt 5486  2c2 5961  intcnt 7661  clsccl 7662   ball cbl 7791  Opencopn 7792  CMetcms 7921

This theorem is referenced by:  bcthlem31 8029

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo