Theorem suppsrlem 5193

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 5150	. . . . . . . 8 |- ~R e. V
2		ecexg 4249	. . . . . . . 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 1526	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (x e. A <-> [<.(w +P. 1P), 1P>.] ~R e. A))
5		breq2 2613	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (0R <R x <-> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
6	4, 5	imbi12d 624	. . . . . . 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 1859	. . . . . 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 1562	. . . . . 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 1804	. . . . . 6 |- w e. V
12	11	mappsrpr 5190	. . . . 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 2060	. . 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 1000	. . . . . . 7 |- (A.x(x e. A -> 0R <R x) -> A.xA.x(x e. A -> 0R <R x))
17		ax-17 968	. . . . . . 7 |- (-. B = (/) -> A.x -. B = (/))
18	16, 17	hbim 1004	. . . . . 6 |- ((A.x(x e. A -> 0R <R x) -> -. B = (/)) -> A.x(A.x(x e. A -> 0R <R x) -> -. B = (/)))
19		ax-4 970	. . . . . . . 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 1526	. . . . . . . . . . . . 13 |- ([<.(w +P. 1P), 1P>.] ~R = x -> ([<.(w +P. 1P), 1P>.] ~R e. A <-> x e. A))
22	21, 9	syl5bb 530	. . . . . . . . . . . 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 2275	. . . . . . . . . . 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 1209	. . . . . . . 8 |- (x e. A -> (E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
28		visset 1804	. . . . . . . . 9 |- x e. V
29	28	map2psrpr 5192	. . . . . . . 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 1060	. . . . 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 2279	. . . 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 951   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  Vcvv 1802   (_ wss 2037  (/)c0 2270  <.cop 2401   class class class wbr 2609  (class class class)co 3948  [cec 4243  P.cnp 4957  1Pc1p 4958   +P. cpp 4959   ~R cer 4964  0Rc0r 4966   <R cltr 4971

This theorem is referenced by:  suppsr 5194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-ltp 5062  df-enr 5138  df-nr 5139  df-ltr 5142  df-0r 5143

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