Theorem suppsrlem 5201

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 5158	. . . . . . . 8 |- ~R e. V
2		ecexg 4255	. . . . . . . 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 1531	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (x e. A <-> [<.(w +P. 1P), 1P>.] ~R e. A))
5		breq2 2618	. . . . . . . 8 |- (x = [<.(w +P. 1P), 1P>.] ~R -> (0R <R x <-> 0R <R [<.(w +P. 1P), 1P>.] ~R ))
6	4, 5	imbi12d 625	. . . . . . 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 1864	. . . . . 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 1567	. . . . . 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 1809	. . . . . 6 |- w e. V
12	11	mappsrpr 5198	. . . . 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 2066	. . 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 1001	. . . . . . 7 |- (A.x(x e. A -> 0R <R x) -> A.xA.x(x e. A -> 0R <R x))
17		ax-17 969	. . . . . . 7 |- (-. B = (/) -> A.x -. B = (/))
18	16, 17	hbim 1005	. . . . . 6 |- ((A.x(x e. A -> 0R <R x) -> -. B = (/)) -> A.x(A.x(x e. A -> 0R <R x) -> -. B = (/)))
19		ax-4 971	. . . . . . . 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 1531	. . . . . . . . . . . . 13 |- ([<.(w +P. 1P), 1P>.] ~R = x -> ([<.(w +P. 1P), 1P>.] ~R e. A <-> x e. A))
22	21, 9	syl5bb 531	. . . . . . . . . . . 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 2281	. . . . . . . . . . 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 1212	. . . . . . . 8 |- (x e. A -> (E.w(w e. P. /\ [<.(w +P. 1P), 1P>.] ~R = x) -> -. B = (/)))
28		visset 1809	. . . . . . . . 9 |- x e. V
29	28	map2psrpr 5200	. . . . . . . 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 1062	. . . . 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 2285	. . . 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 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  Vcvv 1807   (_ wss 2043  (/)c0 2276  <.cop 2407   class class class wbr 2614  (class class class)co 3954  [cec 4249  P.cnp 4965  1Pc1p 4966   +P. cpp 4967   ~R cer 4972  0Rc0r 4974   <R cltr 4979

This theorem is referenced by:  suppsr 5202

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-ltp 5070  df-enr 5146  df-nr 5147  df-ltr 5150  df-0r 5151

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