Theorem suppsrlem 4015

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 ∈ A}

Assertion

Ref	Expression
suppsrlem	⊢ ((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) → (B ⊆ P ∧ ¬ B = ∅))

Distinct variable group(s):   x,w,A   x,B,w

Proof of Theorem suppsrlem

Step	Hyp	Ref	Expression
1		enrex 3972	. . . . . . . 8 ⊢ ~R ∈ V
2		ecexg 3204	. . . . . . . 8 ⊢ ( ~R ∈ V → [⟨(w +P 1P), 1P⟩] ~R ∈ V)
3	1, 2	ax-mp 6	. . . . . . 7 ⊢ [⟨(w +P 1P), 1P⟩] ~R ∈ V
4		eleq1 1149	. . . . . . . 8 ⊢ (x = [⟨(w +P 1P), 1P⟩] ~R → (x ∈ A ↔ [⟨(w +P 1P), 1P⟩] ~R ∈ A))
5		breq2 2066	. . . . . . . 8 ⊢ (x = [⟨(w +P 1P), 1P⟩] ~R → (0R <R x ↔ 0R <R [⟨(w +P 1P), 1P⟩] ~R ))
6	4, 5	imbi12d 474	. . . . . . 7 ⊢ (x = [⟨(w +P 1P), 1P⟩] ~R → ((x ∈ A → 0R <R x) ↔ ([⟨(w +P 1P), 1P⟩] ~R ∈ A → 0R <R [⟨(w +P 1P), 1P⟩] ~R )))
7	3, 6	cla4v 1400	. . . . . 6 ⊢ (∀x(x ∈ A → 0R <R x) → ([⟨(w +P 1P), 1P⟩] ~R ∈ A → 0R <R [⟨(w +P 1P), 1P⟩] ~R ))
8		suppsr.1	. . . . . . 7 ⊢ B = {w∣[⟨(w +P 1P), 1P⟩] ~R ∈ A}
9	8	cleqabi 1176	. . . . . 6 ⊢ (w ∈ B ↔ [⟨(w +P 1P), 1P⟩] ~R ∈ A)
10	7, 9	syl5ib 181	. . . . 5 ⊢ (∀x(x ∈ A → 0R <R x) → (w ∈ B → 0R <R [⟨(w +P 1P), 1P⟩] ~R ))
11		visset 1350	. . . . . 6 ⊢ w ∈ V
12	11	mappsrpr 4012	. . . . 5 ⊢ (0R <R [⟨(w +P 1P), 1P⟩] ~R ↔ w ∈ P)
13	10, 12	syl6ib 185	. . . 4 ⊢ (∀x(x ∈ A → 0R <R x) → (w ∈ B → w ∈ P))
14	13	ssrdv 1509	. . 3 ⊢ (∀x(x ∈ A → 0R <R x) → B ⊆ P)
15	14	adantr 306	. 2 ⊢ ((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) → B ⊆ P)
16		hba1 698	. . . . . . 7 ⊢ (∀x(x ∈ A → 0R <R x) → ∀x∀x(x ∈ A → 0R <R x))
17		ax-17 925	. . . . . . 7 ⊢ (¬ B = ∅ → ∀x ¬ B = ∅)
18	16, 17	hbim 702	. . . . . 6 ⊢ ((∀x(x ∈ A → 0R <R x) → ¬ B = ∅) → ∀x(∀x(x ∈ A → 0R <R x) → ¬ B = ∅))
19		ax-4 673	. . . . . . . 8 ⊢ (∀x(x ∈ A → 0R <R x) → (x ∈ A → 0R <R x))
20	19	com12 13	. . . . . . 7 ⊢ (x ∈ A → (∀x(x ∈ A → 0R <R x) → 0R <R x))
21		eleq1 1149	. . . . . . . . . . . . 13 ⊢ ([⟨(w +P 1P), 1P⟩] ~R = x → ([⟨(w +P 1P), 1P⟩] ~R ∈ A ↔ x ∈ A))
22	21, 9	syl5bb 410	. . . . . . . . . . . 12 ⊢ ([⟨(w +P 1P), 1P⟩] ~R = x → (w ∈ B ↔ x ∈ A))
23	22	biimprcd 138	. . . . . . . . . . 11 ⊢ (x ∈ A → ([⟨(w +P 1P), 1P⟩] ~R = x → w ∈ B))
24		n0i 1712	. . . . . . . . . . 11 ⊢ (w ∈ B → ¬ B = ∅)
25	23, 24	syl6 23	. . . . . . . . . 10 ⊢ (x ∈ A → ([⟨(w +P 1P), 1P⟩] ~R = x → ¬ B = ∅))
26	25	adantld 307	. . . . . . . . 9 ⊢ (x ∈ A → ((w ∈ P ∧ [⟨(w +P 1P), 1P⟩] ~R = x) → ¬ B = ∅))
27	26	19.23adv 954	. . . . . . . 8 ⊢ (x ∈ A → (∃w(w ∈ P ∧ [⟨(w +P 1P), 1P⟩] ~R = x) → ¬ B = ∅))
28		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
29	28	map2psrpr 4014	. . . . . . . 8 ⊢ (0R <R x ↔ ∃w(w ∈ P ∧ [⟨(w +P 1P), 1P⟩] ~R = x))
30	27, 29	syl5ib 181	. . . . . . 7 ⊢ (x ∈ A → (0R <R x → ¬ B = ∅))
31	20, 30	syld 27	. . . . . 6 ⊢ (x ∈ A → (∀x(x ∈ A → 0R <R x) → ¬ B = ∅))
32	18, 31	19.23ai 746	. . . . 5 ⊢ (∃x x ∈ A → (∀x(x ∈ A → 0R <R x) → ¬ B = ∅))
33	32	com12 13	. . . 4 ⊢ (∀x(x ∈ A → 0R <R x) → (∃x x ∈ A → ¬ B = ∅))
34		n0 1714	. . . 4 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
35	33, 34	syl5ib 181	. . 3 ⊢ (∀x(x ∈ A → 0R <R x) → (¬ A = ∅ → ¬ B = ∅))
36	35	imp 277	. 2 ⊢ ((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) → ¬ B = ∅)
37	15, 36	jca 236	1 ⊢ ((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) → (B ⊆ P ∧ ¬ B = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  [cec 3198  Pcnp 3779  1_Pc1p 3780   +_P cpp 3781   ~_R cer 3786  0_Rc0r 3788   <_R cltr 3793

This theorem is referenced by:  suppsr 4016

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-enr 3960  df-nr 3961  df-ltr 3964  df-0r 3965

metamath.org