Theorem rlimrege0 15486

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 4027	. . . 4 ⊢ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ)
5		eldifi 4078	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → 𝑦 ∈ ℂ)
7	6	recld 15101	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) ∈ ℝ)
8		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (ℜ‘𝑤) = (ℜ‘𝑦))
9	8	breq2d 5101	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑦)))
10	9	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ 0 ≤ (ℜ‘𝑤) ↔ ¬ 0 ≤ (ℜ‘𝑦)))
11		notrab 4269	. . . . . . . 8 ⊢ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) = {𝑤 ∈ ℂ ∣ ¬ 0 ≤ (ℜ‘𝑤)}
12	10, 11	elrab2 3645	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) ↔ (𝑦 ∈ ℂ ∧ ¬ 0 ≤ (ℜ‘𝑦)))
13	12	simprbi 496	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → ¬ 0 ≤ (ℜ‘𝑦))
14	13	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ¬ 0 ≤ (ℜ‘𝑦))
15		0re 11114	. . . . . 6 ⊢ 0 ∈ ℝ
16		ltnle 11192	. . . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) < 0)
19	7, 18	negelrpd 12926	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ+)
20	7	renegcld 11544	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ)
21	20	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ∈ ℝ)
22		elrabi 3638	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 𝑧 ∈ ℂ)
23	22	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑧 ∈ ℂ)
24	6	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
25	23, 24	subcld 11472	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (𝑧 − 𝑦) ∈ ℂ)
26	25	recld 15101	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) ∈ ℝ)
27	25	abscld 15346	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
28		0red 11115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ∈ ℝ)
29	23	recld 15101	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑧) ∈ ℝ)
30	24	recld 15101	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑦) ∈ ℝ)
31		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (ℜ‘𝑤) = (ℜ‘𝑧))
32	31	breq2d 5101	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑧)))
33	32	elrab 3642	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝑧 ∈ ℂ ∧ 0 ≤ (ℜ‘𝑧)))
34	33	simprbi 496	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝑧))
35	34	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ≤ (ℜ‘𝑧))
36	28, 29, 30, 35	lesub1dd 11733	. . . . 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 11347	. . . . . 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 15123	. . . . 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 5121	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (ℜ‘(𝑧 − 𝑦)))
41	25	releabsd 15361	. . . 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 11270	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (abs‘(𝑧 − 𝑦)))
43		fveq2 6822	. . . . 5 ⊢ (𝑤 = 𝐵 → (ℜ‘𝑤) = (ℜ‘𝐵))
44	43	breq2d 5101	. . . 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 3644	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
48	1, 2, 4, 19, 42, 47	rlimcld2 15485	. 2 ⊢ (𝜑 → 𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
49		fveq2 6822	. . . . 5 ⊢ (𝑤 = 𝐶 → (ℜ‘𝑤) = (ℜ‘𝐶))
50	49	breq2d 5101	. . . 4 ⊢ (𝑤 = 𝐶 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐶)))
51	50	elrab 3642	. . 3 ⊢ (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝐶 ∈ ℂ ∧ 0 ≤ (ℜ‘𝐶)))
52	51	simprbi 496	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ∖ cdif 3894   ⊆ wss 3897   class class class wbr 5089   ↦ cmpt 5170  ‘cfv 6481  (class class class)co 7346  supcsup 9324  ℂcc 11004  ℝcr 11005  0cc0 11006  +∞cpnf 11143  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147   − cmin 11344  -cneg 11345  ℜcre 15004  abscabs 15141   ⇝_𝑟 crli 15392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-rlim 15396

This theorem is referenced by:  rlimge0  15488