Theorem rlimrege0 15500

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 4030	. . . 4 ⊢ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ)
5		eldifi 4081	. . . . . 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 15115	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) ∈ ℝ)
8		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (ℜ‘𝑤) = (ℜ‘𝑦))
9	8	breq2d 5108	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑦)))
10	9	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ 0 ≤ (ℜ‘𝑤) ↔ ¬ 0 ≤ (ℜ‘𝑦)))
11		notrab 4272	. . . . . . . 8 ⊢ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) = {𝑤 ∈ ℂ ∣ ¬ 0 ≤ (ℜ‘𝑤)}
12	10, 11	elrab2 3647	. . . . . . 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 11132	. . . . . 6 ⊢ 0 ∈ ℝ
16		ltnle 11210	. . . . . 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 12939	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ+)
20	7	renegcld 11562	. . . . 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 3640	. . . . . . 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 11490	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (𝑧 − 𝑦) ∈ ℂ)
26	25	recld 15115	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧 − 𝑦)) ∈ ℝ)
27	25	abscld 15360	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
28		0red 11133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ∈ ℝ)
29	23	recld 15115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑧) ∈ ℝ)
30	24	recld 15115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑦) ∈ ℝ)
31		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (ℜ‘𝑤) = (ℜ‘𝑧))
32	31	breq2d 5108	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑧)))
33	32	elrab 3644	. . . . . . . 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 11751	. . . . 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 11365	. . . . . 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 15137	. . . . 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 5128	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (ℜ‘(𝑧 − 𝑦)))
41	25	releabsd 15375	. . . 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 11288	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (abs‘(𝑧 − 𝑦)))
43		fveq2 6832	. . . . 5 ⊢ (𝑤 = 𝐵 → (ℜ‘𝑤) = (ℜ‘𝐵))
44	43	breq2d 5108	. . . 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 3646	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
48	1, 2, 4, 19, 42, 47	rlimcld2 15499	. 2 ⊢ (𝜑 → 𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
49		fveq2 6832	. . . . 5 ⊢ (𝑤 = 𝐶 → (ℜ‘𝑤) = (ℜ‘𝐶))
50	49	breq2d 5108	. . . 4 ⊢ (𝑤 = 𝐶 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐶)))
51	50	elrab 3644	. . 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 2113  {crab 3397   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5096   ↦ cmpt 5177  ‘cfv 6490  (class class class)co 7356  supcsup 9341  ℂcc 11022  ℝcr 11023  0cc0 11024  +∞cpnf 11161  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165   − cmin 11362  -cneg 11363  ℜcre 15018  abscabs 15155   ⇝_𝑟 crli 15406

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-sup 9343  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-seq 13923  df-exp 13983  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-rlim 15410

This theorem is referenced by:  rlimge0  15502