Theorem rlimrege0 15629

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 4093	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
6	5	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → 𝑦 ∈ ℂ)
7	6	recld 15244	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) ∈ ℝ)
8		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (ℜ‘𝑤) = (ℜ‘𝑦))
9	8	breq2d 5125	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑦)))
10	9	notbid 321	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ 0 ≤ (ℜ‘𝑤) ↔ ¬ 0 ≤ (ℜ‘𝑦)))
11		notrab 4283	. . . . . . . 8 ⊢ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) = {𝑤 ∈ ℂ ∣ ¬ 0 ≤ (ℜ‘𝑤)}
12	10, 11	elrab2 3663	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) ↔ (𝑦 ∈ ℂ ∧ ¬ 0 ≤ (ℜ‘𝑦)))
13	12	simprbi 502	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → ¬ 0 ≤ (ℜ‘𝑦))
14	13	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ¬ 0 ≤ (ℜ‘𝑦))
15		0re 11209	. . . . . 6 ⊢ 0 ∈ ℝ
16		ltnle 11288	. . . . . 6 ⊢ (((ℜ‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
17	7, 15, 16	sylancl 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
18	14, 17	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) < 0)
19	7, 18	negelrpd 13051	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ+)
20	7	renegcld 11640	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ)
21	20	adantr 485	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ∈ ℝ)
22		elrabi 3655	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 𝑧 ∈ ℂ)
23	22	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑧 ∈ ℂ)
24	6	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
25	23, 24	subcld 11568	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (𝑧 − 𝑦) ∈ ℂ)
26	25	recld 15244	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) ∈ ℝ)
27	25	abscld 15489	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
28		0red 11210	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ∈ ℝ)
29	23	recld 15244	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑧) ∈ ℝ)
30	24	recld 15244	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑦) ∈ ℝ)
31		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (ℜ‘𝑤) = (ℜ‘𝑧))
32	31	breq2d 5125	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑧)))
33	32	elrab 3659	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝑧 ∈ ℂ ∧ 0 ≤ (ℜ‘𝑧)))
34	33	simprbi 502	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝑧))
35	34	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ≤ (ℜ‘𝑧))
36	28, 29, 30, 35	lesub1dd 11829	. . . . 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 11443	. . . . . 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 15266	. . . . 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 5147	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (ℜ‘(𝑧 − 𝑦)))
41	25	releabsd 15504	. . . 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 11366	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (abs‘(𝑧 − 𝑦)))
43		fveq2 6882	. . . . 5 ⊢ (𝑤 = 𝐵 → (ℜ‘𝑤) = (ℜ‘𝐵))
44	43	breq2d 5125	. . . 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 3661	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
48	1, 2, 4, 19, 42, 47	rlimcld2 15628	. 2 ⊢ (𝜑 → 𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
49		fveq2 6882	. . . . 5 ⊢ (𝑤 = 𝐶 → (ℜ‘𝑤) = (ℜ‘𝐶))
50	49	breq2d 5125	. . . 4 ⊢ (𝑤 = 𝐶 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐶)))
51	50	elrab 3659	. . 3 ⊢ (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝐶 ∈ ℂ ∧ 0 ≤ (ℜ‘𝐶)))
52	51	simprbi 502	. 2 ⊢ (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝐶))
53	48, 52	syl 18	1 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423   ∖ cdif 3910   ⊆ wss 3913   class class class wbr 5113   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  supcsup 9399  ℂcc 11097  ℝcr 11098  0cc0 11099  +∞cpnf 11239  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243   − cmin 11440  -cneg 11441  ℜcre 15147  abscabs 15284   ⇝_𝑟 crli 15535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-seq 14037  df-exp 14097  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-rlim 15539

This theorem is referenced by:  rlimge0  15631