Theorem suppsrlem 5233

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 5190	. . . . . . . 8 |- ~R e. V
2		ecexg 4271	. . . . . . . 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 1537	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (x e. A <-> [<.(w +P. 1P), 1P>.] ~R e. A))
5		breq2 2628	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (0R <R x <-> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
6	4, 5	imbi12d 628	. . . . . . 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 1871	. . . . . 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 1573	. . . . . 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 1816	. . . . . 6 |- w e. V
12	11	mappsrpr 5230	. . . . 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 2073	. . 3 |- (A.x(x e. A -> 0R <R x) -> B (_ P.)
15	14	adantr 391	. 2 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> B (_ P.)
16		hba1 1005	. . . . . . 7 |- (A.x(x e. A -> 0R <R x) -> A.xA.x(x e. A -> 0R <R x))
17		ax-17 973	. . . . . . 7 |- (-. B = (/) -> A.x -. B = (/))
18	16, 17	hbim 1009	. . . . . 6 |- ((A.x(x e. A -> 0R <R x) -> -. B = (/)) -> A.x(A.x(x e. A -> 0R <R x) -> -. B = (/)))
19		ax-4 975	. . . . . . . 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 1537	. . . . . . . . . . . . 13 |- ([<.(w +P. 1P), 1P>.] ~R = x -> ([<.(w +P. 1P), 1P>.] ~R e. A <-> x e. A))
22	21, 9	syl5bb 534	. . . . . . . . . . . 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 2288	. . . . . . . . . . 11 |- (w e. B -> -. B = (/))
25	23, 24	syl6 22	. . . . . . . . . 10 |- (x e. A -> ([<.(w +P. 1P), 1P>.] ~R = x -> -. B = (/)))
26	25	adantld 392	. . . . . . . . 9 |- (x e. A -> ((w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
27	26	19.23adv 1216	. . . . . . . 8 |- (x e. A -> (E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
28		visset 1816	. . . . . . . . 9 |- x e. V
29	28	map2psrpr 5232	. . . . . . . 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 1066	. . . . 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 2293	. . . 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 956   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  Vcvv 1814   (_ wss 2050  (/)c0 2283  <.cop 2415   class class class wbr 2624  (class class class)co 3969  [cec 4265  P.cnp 4997  1Pc1p 4998   +P. cpp 4999   ~R cer 5004  0Rc0r 5006   <R cltr 5011

This theorem is referenced by:  suppsr 5234

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-ltp 5102  df-enr 5178  df-nr 5179  df-ltr 5182  df-0r 5183

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