Theorem suppsrlem 6373

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 C_ P. /\ -. B = (/)))

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

Proof of Theorem suppsrlem

Step	Hyp	Ref	Expression
1		enrex 6330	. . . . . . . 8 |- ~R e. _V
2		ecexg 5322	. . . . . . . 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 1957	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (x e. A <-> [<.(w +P. 1P), 1P>.] ~R e. A))
5		breq2 3342	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (0R <R x <-> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
6	4, 5	imbi12d 688	. . . . . . 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 2370	. . . . . 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 2001	. . . . . 6 |- (w e. B <-> [<.(w +P. 1P), 1P>.] ~R e. A)
10	7, 9	syl5ib 223	. . . . 5 |- (A.x(x e. A -> 0R <R x) -> (w e. B -> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
11		visset 2295	. . . . . 6 |- w e. _V
12	11	mappsrpr 6370	. . . . 5 |- (0R <R [<.(w +P. 1P), 1P>.] ~R <-> w e. P.)
13	10, 12	syl6ib 229	. . . 4 |- (A.x(x e. A -> 0R <R x) -> (w e. B -> w e. P.))
14	13	ssrdv 2622	. . 3 |- (A.x(x e. A -> 0R <R x) -> B C_ P.)
15	14	adantr 425	. 2 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> B C_ P.)
16		hba1 1350	. . . . . . 7 |- (A.x(x e. A -> 0R <R x) -> A.xA.x(x e. A -> 0R <R x))
17		ax-17 1317	. . . . . . 7 |- (-. B = (/) -> A.x -. B = (/))
18	16, 17	hbim 1354	. . . . . 6 |- ((A.x(x e. A -> 0R <R x) -> -. B = (/)) -> A.x(A.x(x e. A -> 0R <R x) -> -. B = (/)))
19		ax-4 1319	. . . . . . . 8 |- (A.x(x e. A -> 0R <R x) -> (x e. A -> 0R <R x))
20	19	com12 14	. . . . . . 7 |- (x e. A -> (A.x(x e. A -> 0R <R x) -> 0R <R x))
21		eleq1 1957	. . . . . . . . . . . . 13 |- ([<.(w +P. 1P), 1P>.] ~R = x -> ([<.(w +P. 1P), 1P>.] ~R e. A <-> x e. A))
22	21, 9	syl5bb 591	. . . . . . . . . . . 12 |- ([<.(w +P. 1P), 1P>.] ~R = x -> (w e. B <-> x e. A))
23	22	biimprcd 173	. . . . . . . . . . 11 |- (x e. A -> ([<.(w +P. 1P), 1P>.] ~R = x -> w e. B))
24		n0i 2880	. . . . . . . . . . 11 |- (w e. B -> -. B = (/))
25	23, 24	syl6 25	. . . . . . . . . 10 |- (x e. A -> ([<.(w +P. 1P), 1P>.] ~R = x -> -. B = (/)))
26	25	adantld 426	. . . . . . . . 9 |- (x e. A -> ((w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
27	26	19.23adv 1584	. . . . . . . 8 |- (x e. A -> (E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
28		visset 2295	. . . . . . . . 9 |- x e. _V
29	28	map2psrpr 6372	. . . . . . . 8 |- (0R <R x <-> E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x))
30	27, 29	syl5ib 223	. . . . . . 7 |- (x e. A -> (0R <R x -> -. B = (/)))
31	20, 30	syld 30	. . . . . 6 |- (x e. A -> (A.x(x e. A -> 0R <R x) -> -. B = (/)))
32	18, 31	19.23ai 1412	. . . . 5 |- (E.x x e. A -> (A.x(x e. A -> 0R <R x) -> -. B = (/)))
33	32	com12 14	. . . 4 |- (A.x(x e. A -> 0R <R x) -> (E.x x e. A -> -. B = (/)))
34		neq0 2885	. . . 4 |- (-. A = (/) <-> E.x x e. A)
35	33, 34	syl5ib 223	. . 3 |- (A.x(x e. A -> 0R <R x) -> (-. A = (/) -> -. B = (/)))
36	35	imp 377	. 2 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> -. B = (/))
37	15, 36	jca 310	1 |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B C_ P. /\ -. B = (/)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292   C_ wss 2593  (/)c0 2875  <.cop 3046   class class class wbr 3338  (class class class)co 4884  [cec 5316  P.cnp 6137  1Pc1p 6138   +P. cpp 6139   ~R cer 6144  0Rc0r 6146   <R cltr 6151

This theorem is referenced by:  suppsr 6374

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323

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