Theorem bcthlem31 8029

Description: Lemma for bcth 8032. Eliminate the antecedents involving sequence 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
bcthlem31	|- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 1)))) /\ (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ Q e. A)) -> -. U.ran M = X)

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

Proof of Theorem bcthlem31

Step	Hyp	Ref	Expression
1		bcthlem29.1	. . . . 5 |- D e. CMet
2		bcthlem29.3	. . . . 5 |- X = dom dom D
3		bcthlem29.4	. . . . 5 |- J = (Open` D)
4		bcthlem29.5	. . . . 5 |- M:NN-->P~X
5		bcthlem29.6	. . . . 5 |- 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))})}
6		bcthlem29.7	. . . . 5 |- A = (X X. {x e. RR | 0 < x})
7		bcthlem29.8	. . . . 5 |- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))
8	1, 2, 3, 4, 5, 6, 7	bcthlem30 8028	. . . 4 |- ((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))))
9	8	adantl 388	. . 3 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 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))))
10		3anass 779	. . . 4 |- ((g:NN-->A /\ (g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) <-> (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
11	10	exbii 1051	. . 3 |- (E.g(g:NN-->A /\ (g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) <-> E.g(g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
12	9, 11	sylib 198	. 2 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 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)))))
13	1, 2, 3, 4, 5, 6, 7	bcthlem29 8027	. . . . 5 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> -. U.ran M = X)
14	13	ex 373	. . . 4 |- (((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 1)))) -> ((g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) -> -. U.ran M = X))
15	14	19.23adv 1214	. . 3 |- (((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 1)))) -> (E.g(g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) -> -. U.ran M = X))
16	15	adantr 389	. 2 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 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)))) -> -. U.ran M = X))
17	12, 16	mpd 26	1 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) (_ (X \ ((cls` J)` (M` 1)))) /\ (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ Q e. A)) -> -. U.ran M = X)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  A.wral 1645  {crab 1648   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280  P~cpw 2401  U.cuni 2503   class class class wbr 2619  {copab 2666   X. cxp 3168  dom cdm 3170  ran crn 3171  -->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:  bcthlem32 8030

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 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-n0 6100  df-z 6136  df-fl 6224  df-seq1 6308  df-uz 6418  df-exp 6569  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665  df-met 7793  df-bl 7795  df-opn 7796  df-lm 7922  df-cau 7923  df-cmet 7924

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