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Theorem psercnlem1 26424

Description: Lemma for psercn 26425. (Contributed by Mario Carneiro, 18-Mar-2015.)

**Hypotheses**
Ref	Expression
pserf.g	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
pserf.f	⊢ 𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ₀ ((𝐺‘𝑦)‘𝑗))
pserf.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
pserf.r	⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
psercn.s	⊢ 𝑆 = (^◡abs “ (0[,)𝑅))
psercn.m	⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))

**Assertion**
Ref	Expression
psercnlem1	⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ⁺ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))

Distinct variable groups: 𝑗,𝑎,𝑛,𝑟,𝑥,𝑦,𝐴 𝑗,𝑀,𝑦 𝑗,𝐺,𝑟,𝑦 𝑆,𝑎,𝑗,𝑦 𝐹,𝑎 𝜑,𝑎,𝑗,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑛,𝑟) 𝑅(𝑥,𝑦,𝑗,𝑛,𝑟,𝑎) 𝑆(𝑥,𝑛,𝑟) 𝐹(𝑥,𝑦,𝑗,𝑛,𝑟) 𝐺(𝑥,𝑛,𝑎) 𝑀(𝑥,𝑛,𝑟,𝑎)

**Proof of Theorem psercnlem1**
Step	Hyp	Ref	Expression
1		psercn.m	. . . 4 ⊢ 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))
2		psercn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (^◡abs “ (0[,)𝑅))
3		cnvimass 6082	. . . . . . . . . . . 12 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
4		absf 15359	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
5	4	fdmi 6728	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
6	3, 5	sseqtri 4014	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
7	2, 6	eqsstri 4012	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	sselda 3965	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
10	9	abscld 15458	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
11		readdcl 11221	. . . . . . 7 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 𝑅) ∈ ℝ)
12	10, 11	sylan 580	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 𝑅) ∈ ℝ)
13	12	rehalfcld 12497	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) ∈ ℝ)
14		peano2re 11417	. . . . . . 7 ⊢ ((abs‘𝑎) ∈ ℝ → ((abs‘𝑎) + 1) ∈ ℝ)
15	10, 14	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ)
16	15	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) ∈ ℝ)
17	13, 16	ifclda 4543	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ∈ ℝ)
18	1, 17	eqeltrid 2837	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
19		0re 11246	. . . . 5 ⊢ 0 ∈ ℝ
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ)
21	9	absge0d 15466	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎))
22		breq2 5129	. . . . . 6 ⊢ ((((abs‘𝑎) + 𝑅) / 2) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
23		breq2 5129	. . . . . 6 ⊢ (((abs‘𝑎) + 1) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < ((abs‘𝑎) + 1) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
24		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
25	24, 2	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (^◡abs “ (0[,)𝑅)))
26		ffn 6717	. . . . . . . . . . . . 13 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27		elpreima 7059	. . . . . . . . . . . . 13 ⊢ (abs Fn ℂ → (𝑎 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅))))
28	4, 26, 27	mp2b 10	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
29	25, 28	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
30	29	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
31		iccssxr 13453	. . . . . . . . . . . 12 ⊢ (0[,]+∞) ⊆ ℝ^*
32		pserf.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
33		pserf.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
34		pserf.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
35	32, 33, 34	radcnvcl 26415	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
37	31, 36	sselid 3963	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ^*)
38		elico2 13434	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
39	19, 37, 38	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
40	30, 39	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
41	40	simp3d 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
42	41	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
43		avglt1 12488	. . . . . . . 8 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
44	10, 43	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
45	42, 44	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
46	10	ltp1d 12181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
47	46	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
48	22, 23, 45, 47	ifbothda 4546	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
49	48, 1	breqtrrdi 5167	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
50	20, 10, 18, 21, 49	lelttrd 11402	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < 𝑀)
51	18, 50	elrpd 13057	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ⁺)
52		breq1 5128	. . . 4 ⊢ ((((abs‘𝑎) + 𝑅) / 2) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((((abs‘𝑎) + 𝑅) / 2) < 𝑅 ↔ if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅))
53		breq1 5128	. . . 4 ⊢ (((abs‘𝑎) + 1) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → (((abs‘𝑎) + 1) < 𝑅 ↔ if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅))
54		avglt2 12489	. . . . . 6 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (((abs‘𝑎) + 𝑅) / 2) < 𝑅))
55	10, 54	sylan 580	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (((abs‘𝑎) + 𝑅) / 2) < 𝑅))
56	42, 55	mpbid 232	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) < 𝑅)
57	15	rexrd 11294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ^*)
58	37, 57	xrlenltd 11310	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) ↔ ¬ ((abs‘𝑎) + 1) < 𝑅))
59		0xr 11291	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
60		pnfxr 11298	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
61		elicc1 13414	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^ ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)))
62	59, 60, 61	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
63	35, 62	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
64	63	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑅)
65	64	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ 𝑅)
66		ge0gtmnf 13197	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → -∞ < 𝑅)
67	37, 65, 66	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → -∞ < 𝑅)
68		xrre 13194	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ (-∞ < 𝑅 ∧ 𝑅 ≤ ((abs‘𝑎) + 1))) → 𝑅 ∈ ℝ)
69	68	expr 456	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ -∞ < 𝑅) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
70	37, 15, 67, 69	syl21anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
71	58, 70	sylbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (¬ ((abs‘𝑎) + 1) < 𝑅 → 𝑅 ∈ ℝ))
72	71	con1d 145	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (¬ 𝑅 ∈ ℝ → ((abs‘𝑎) + 1) < 𝑅))
73	72	imp 406	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) < 𝑅)
74	52, 53, 56, 73	ifbothda 4546	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅)
75	1, 74	eqbrtrid 5160	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
76	51, 49, 75	3jca 1128	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ⁺ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3420 ⊆ wss 3933 ifcif 4507 class class class wbr 5125 ↦ cmpt 5207 ^◡ccnv 5666 dom cdm 5667 “ cima 5670 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 supcsup 9463 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 +∞cpnf 11275 -∞cmnf 11276 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 / cdiv 11903 2c2 12304 ℕ₀cn0 12510 ℝ⁺crp 13017 [,)cico 13372 [,]cicc 13373 seqcseq 14025 ↑cexp 14085 abscabs 15256 ⇝ cli 15503 Σcsu 15705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-er 8728 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-div 11904 df-nn 12250 df-2 12312 df-3 12313 df-n0 12511 df-z 12598 df-uz 12862 df-rp 13018 df-ico 13376 df-icc 13377 df-fz 13531 df-seq 14026 df-exp 14086 df-cj 15121 df-re 15122 df-im 15123 df-sqrt 15257 df-abs 15258 df-clim 15507

This theorem is referenced by: psercn 26425 pserdvlem1 26426 pserdvlem2 26427 pserdv 26428

W3C validator