Theorem suppsrlem 5221

Description: Mapping of non-empty subset from positive reals to positive signed reals.

Hypothesis

Ref	Expression
suppsr.1	|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}

Assertion

Ref	Expression
suppsrlem	|- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B (_ P. /\ -. B = (/)))

Distinct variable groups:   x,w,A   x,B,w

Proof of Theorem suppsrlem

Step	Hyp	Ref	Expression
1		enrex 5178	. . . . . . . 8 |- ~R e. V
2		ecexg 4265	. . . . . . . 8 |- ( ~R e. V -> [<.(w +P. 1P), 1P>.] ~R e. V)
3	1, 2	ax-mp 7	. . . . . . 7 |- [<.(w +P. 1P), 1P>.] ~R e. V
4		eleq1 1534	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (x e. A <-> [<.(w +P. 1P), 1P>.] ~R e. A))
5		breq2 2623	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (0R <R x <-> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
6	4, 5	imbi12d 626	. . . . . . 7 |- (x = [<.(w +P. 1P), 1P>.] ~R -> ((x e. A -> 0R <R x) <-> ([<.(w +P. 1P), 1P>.] ~R e. A -> 0R <R [<.(w +P. 1P), 1P>.] ~R )))
7	3, 6	cla4v 1868	. . . . . 6 |- (A.x(x e. A -> 0R <R x) -> ([<.(w +P. 1P), 1P>.] ~R e. A -> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
8		suppsr.1	. . . . . . 7 |- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}
9	8	abeq2i 1570	. . . . . 6 |- (w e. B <-> [<.(w +P. 1P), 1P>.] ~R e. A)
10	7, 9	syl5ib 206	. . . . 5 |- (A.x(x e. A -> 0R <R x) -> (w e. B -> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
11		visset 1813	. . . . . 6 |- w e. V
12	11	mappsrpr 5218	. . . . 5 |- (0R <R [<.(w +P. 1P), 1P>.] ~R <-> w e. P.)
13	10, 12	syl6ib 212	. . . 4 |- (A.x(x e. A -> 0R <R x) -> (w e. B -> w e. P.))
14	13	ssrdv 2070	. . 3 |- (A.x(x e. A -> 0R <R x) -> B (_ P.)
15	14	adantr 389	. 2 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> B (_ P.)
16		hba1 1003	. . . . . . 7 |- (A.x(x e. A -> 0R <R x) -> A.xA.x(x e. A -> 0R <R x))
17		ax-17 971	. . . . . . 7 |- (-. B = (/) -> A.x -. B = (/))
18	16, 17	hbim 1007	. . . . . 6 |- ((A.x(x e. A -> 0R <R x) -> -. B = (/)) -> A.x(A.x(x e. A -> 0R <R x) -> -. B = (/)))
19		ax-4 973	. . . . . . . 8 |- (A.x(x e. A -> 0R <R x) -> (x e. A -> 0R <R x))
20	19	com12 11	. . . . . . 7 |- (x e. A -> (A.x(x e. A -> 0R <R x) -> 0R <R x))
21		eleq1 1534	. . . . . . . . . . . . 13 |- ([<.(w +P. 1P), 1P>.] ~R = x -> ([<.(w +P. 1P), 1P>.] ~R e. A <-> x e. A))
22	21, 9	syl5bb 532	. . . . . . . . . . . 12 |- ([<.(w +P. 1P), 1P>.] ~R = x -> (w e. B <-> x e. A))
23	22	biimprcd 156	. . . . . . . . . . 11 |- (x e. A -> ([<.(w +P. 1P), 1P>.] ~R = x -> w e. B))
24		n0i 2285	. . . . . . . . . . 11 |- (w e. B -> -. B = (/))
25	23, 24	syl6 22	. . . . . . . . . 10 |- (x e. A -> ([<.(w +P. 1P), 1P>.] ~R = x -> -. B = (/)))
26	25	adantld 390	. . . . . . . . 9 |- (x e. A -> ((w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
27	26	19.23adv 1214	. . . . . . . 8 |- (x e. A -> (E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
28		visset 1813	. . . . . . . . 9 |- x e. V
29	28	map2psrpr 5220	. . . . . . . 8 |- (0R <R x <-> E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x))
30	27, 29	syl5ib 206	. . . . . . 7 |- (x e. A -> (0R <R x -> -. B = (/)))
31	20, 30	syld 27	. . . . . 6 |- (x e. A -> (A.x(x e. A -> 0R <R x) -> -. B = (/)))
32	18, 31	19.23ai 1064	. . . . 5 |- (E.x x e. A -> (A.x(x e. A -> 0R <R x) -> -. B = (/)))
33	32	com12 11	. . . 4 |- (A.x(x e. A -> 0R <R x) -> (E.x x e. A -> -. B = (/)))
34		n0 2289	. . . 4 |- (-. A = (/) <-> E.x x e. A)
35	33, 34	syl5ib 206	. . 3 |- (A.x(x e. A -> 0R <R x) -> (-. A = (/) -> -. B = (/)))
36	35	imp 350	. 2 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> -. B = (/))
37	15, 36	jca 288	1 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B (_ P. /\ -. B = (/)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811   (_ wss 2047  (/)c0 2280  <.cop 2411   class class class wbr 2619  (class class class)co 3963  [cec 4259  P.cnp 4985  1Pc1p 4986   +P. cpp 4987   ~R cer 4992  0Rc0r 4994   <R cltr 4999

This theorem is referenced by:  suppsr 5222

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

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-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-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-fv 3198  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-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-ltp 5090  df-enr 5166  df-nr 5167  df-ltr 5170  df-0r 5171

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