Theorem bcthlem32 8030

Description: Lemma for bcth 8032. Eliminate hypotheses no longer needed.

Hypotheses

Ref	Expression
bcthlem33.1	|- D e. CMet
bcthlem33.3	|- X = dom dom D
bcthlem33.4	|- J = (Open` D)
bcthlem33.5	|- M:NN-->P~X

Assertion

Ref	Expression
bcthlem32	|- (((s < (1 / 2) /\ (q( ball ` D)s) (_ (X \ ((cls` J)` (M` 1)))) /\ (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ (q e. X /\ (s e. RR /\ 0 < s)))) -> -. U.ran M = X)

Distinct variable groups:   D,j   j,J   j,M   j,X

Proof of Theorem bcthlem32

Step	Hyp	Ref	Expression
1		bcthlem33.1	. . 3 |- D e. CMet
2		bcthlem33.3	. . 3 |- X = dom dom D
3		bcthlem33.4	. . 3 |- J = (Open` D)
4		bcthlem33.5	. . 3 |- M:NN-->P~X
5		eqid 1475	. . 3 |- {<.<.j, y>., z>. | ((j e. NN /\ y e. (X X. {x e. RR | 0 < x})) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))))})} = {<.<.j, y>., z>. | ((j e. NN /\ y e. (X X. {x e. RR | 0 < x})) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) (_ ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))))})}
6		eqid 1475	. . 3 |- (X X. {x e. RR | 0 < x}) = (X X. {x e. RR | 0 < x})
7		eqid 1475	. . 3 |- ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2))) = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))
8	1, 2, 3, 4, 5, 6, 7	bcthlem31 8029	. 2 |- ((((2nd` <.q, s>.) < (1 / 2) /\ ((1st` <.q, s>.)( ball ` D)(2nd` <.q, s>.)) (_ (X \ ((cls` J)` (M` 1)))) /\ (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ <.q, s>. e. (X X. {x e. RR | 0 < x}))) -> -. U.ran M = X)
9		visset 1813	. . . . 5 |- q e. V
10		visset 1813	. . . . 5 |- s e. V
11	9, 10	op2nd 4086	. . . 4 |- (2nd` <.q, s>.) = s
12	11	breq1i 2626	. . 3 |- ((2nd` <.q, s>.) < (1 / 2) <-> s < (1 / 2))
13	9	op1st 4085	. . . . 5 |- (1st` <.q, s>.) = q
14	13, 11	opreq12i 3973	. . . 4 |- ((1st` <.q, s>.)( ball ` D)(2nd` <.q, s>.)) = (q( ball ` D)s)
15	14	sseq1i 2085	. . 3 |- (((1st` <.q, s>.)( ball ` D)(2nd` <.q, s>.)) (_ (X \ ((cls` J)` (M` 1))) <-> (q( ball ` D)s) (_ (X \ ((cls` J)` (M` 1))))
16	12, 15	anbi12i 482	. 2 |- (((2nd` <.q, s>.) < (1 / 2) /\ ((1st` <.q, s>.)( ball ` D)(2nd` <.q, s>.)) (_ (X \ ((cls` J)` (M` 1)))) <-> (s < (1 / 2) /\ (q( ball ` D)s) (_ (X \ ((cls` J)` (M` 1)))))
17	10	opelxp 3214	. . . 4 |- (<.q, s>. e. (X X. {x e. RR | 0 < x}) <-> (q e. X /\ s e. {x e. RR | 0 < x}))
18		breq2 2623	. . . . . 6 |- (x = s -> (0 < x <-> 0 < s))
19	18	elrab 1905	. . . . 5 |- (s e. {x e. RR | 0 < x} <-> (s e. RR /\ 0 < s))
20	19	anbi2i 480	. . . 4 |- ((q e. X /\ s e. {x e. RR | 0 < x}) <-> (q e. X /\ (s e. RR /\ 0 < s)))
21	17, 20	bitr 173	. . 3 |- (<.q, s>. e. (X X. {x e. RR | 0 < x}) <-> (q e. X /\ (s e. RR /\ 0 < s)))
22	21	anbi2i 480	. 2 |- ((A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ <.q, s>. e. (X X. {x e. RR | 0 < x})) <-> (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ (q e. X /\ (s e. RR /\ 0 < s))))
23	8, 16, 22	syl2anbr 456	1 |- (((s < (1 / 2) /\ (q( ball ` D)s) (_ (X \ ((cls` J)` (M` 1)))) /\ (A.j e. NN ((int` J)` ((cls` J)` (M` j))) = (/) /\ (q e. X /\ (s e. RR /\ 0 < s)))) -> -. U.ran M = X)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280  P~cpw 2401  <.cop 2411  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   / cdiv 5294  NNcn 5296   < clt 5486  2c2 5961  intcnt 7661  clsccl 7662   ball cbl 7791  Opencopn 7792  CMetcms 7921

This theorem is referenced by:  bcthlem33 8031

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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