Theorem rlimrege0 15473

Description: The limit of a sequence of complex numbers with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)

Hypotheses

Ref	Expression
rlimcld2.1	⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
rlimcld2.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)
rlimrege0.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
rlimrege0.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (ℜ‘𝐵))

Assertion

Ref	Expression
rlimrege0	⊢ (𝜑 → 0 ≤ (ℜ‘𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rlimrege0

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimcld2.1	. . 3 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
2		rlimcld2.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)
3		ssrab2 4042	. . . 4 ⊢ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ)
5		eldifi 4091	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
6	5	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → 𝑦 ∈ ℂ)
7	6	recld 15091	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) ∈ ℝ)
8		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (ℜ‘𝑤) = (ℜ‘𝑦))
9	8	breq2d 5122	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑦)))
10	9	notbid 317	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ 0 ≤ (ℜ‘𝑤) ↔ ¬ 0 ≤ (ℜ‘𝑦)))
11		notrab 4276	. . . . . . . 8 ⊢ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) = {𝑤 ∈ ℂ ∣ ¬ 0 ≤ (ℜ‘𝑤)}
12	10, 11	elrab2 3651	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) ↔ (𝑦 ∈ ℂ ∧ ¬ 0 ≤ (ℜ‘𝑦)))
13	12	simprbi 497	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → ¬ 0 ≤ (ℜ‘𝑦))
14	13	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ¬ 0 ≤ (ℜ‘𝑦))
15		0re 11166	. . . . . 6 ⊢ 0 ∈ ℝ
16		ltnle 11243	. . . . . 6 ⊢ (((ℜ‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
17	7, 15, 16	sylancl 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
18	14, 17	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) < 0)
19	7, 18	negelrpd 12958	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ+)
20	7	renegcld 11591	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ)
21	20	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ∈ ℝ)
22		elrabi 3642	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 𝑧 ∈ ℂ)
23	22	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑧 ∈ ℂ)
24	6	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
25	23, 24	subcld 11521	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (𝑧 − 𝑦) ∈ ℂ)
26	25	recld 15091	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) ∈ ℝ)
27	25	abscld 15333	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
28		0red 11167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ∈ ℝ)
29	23	recld 15091	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑧) ∈ ℝ)
30	24	recld 15091	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑦) ∈ ℝ)
31		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (ℜ‘𝑤) = (ℜ‘𝑧))
32	31	breq2d 5122	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑧)))
33	32	elrab 3648	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝑧 ∈ ℂ ∧ 0 ≤ (ℜ‘𝑧)))
34	33	simprbi 497	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝑧))
35	34	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ≤ (ℜ‘𝑧))
36	28, 29, 30, 35	lesub1dd 11780	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (0 − (ℜ‘𝑦)) ≤ ((ℜ‘𝑧) − (ℜ‘𝑦)))
37		df-neg 11397	. . . . . 6 ⊢ -(ℜ‘𝑦) = (0 − (ℜ‘𝑦))
38	37	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) = (0 − (ℜ‘𝑦)))
39	23, 24	resubd 15113	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) = ((ℜ‘𝑧) − (ℜ‘𝑦)))
40	36, 38, 39	3brtr4d 5142	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (ℜ‘(𝑧 − 𝑦)))
41	25	releabsd 15348	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
42	21, 26, 27, 40, 41	letrd 11321	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (abs‘(𝑧 − 𝑦)))
43		fveq2 6847	. . . . 5 ⊢ (𝑤 = 𝐵 → (ℜ‘𝑤) = (ℜ‘𝐵))
44	43	breq2d 5122	. . . 4 ⊢ (𝑤 = 𝐵 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐵)))
45		rlimrege0.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
46		rlimrege0.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (ℜ‘𝐵))
47	44, 45, 46	elrabd 3650	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
48	1, 2, 4, 19, 42, 47	rlimcld2 15472	. 2 ⊢ (𝜑 → 𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
49		fveq2 6847	. . . . 5 ⊢ (𝑤 = 𝐶 → (ℜ‘𝑤) = (ℜ‘𝐶))
50	49	breq2d 5122	. . . 4 ⊢ (𝑤 = 𝐶 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐶)))
51	50	elrab 3648	. . 3 ⊢ (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝐶 ∈ ℂ ∧ 0 ≤ (ℜ‘𝐶)))
52	51	simprbi 497	. 2 ⊢ (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝐶))
53	48, 52	syl 17	1 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3405   ∖ cdif 3910   ⊆ wss 3913   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6501  (class class class)co 7362  supcsup 9385  ℂcc 11058  ℝcr 11059  0cc0 11060  +∞cpnf 11195  ℝ^*cxr 11197   < clt 11198   ≤ cle 11199   − cmin 11394  -cneg 11395  ℜcre 14994  abscabs 15131   ⇝_𝑟 crli 15379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9387  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-seq 13917  df-exp 13978  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-rlim 15383

This theorem is referenced by:  rlimge0  15475