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Theorem psercnlem1 25582

Description: Lemma for psercn 25583. (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 5988	. . . . . . . . . . . 12 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
4		absf 15047	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
5	4	fdmi 6610	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
6	3, 5	sseqtri 3962	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
7	2, 6	eqsstri 3960	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	sselda 3926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
10	9	abscld 15146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
11		readdcl 10955	. . . . . . 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 12220	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) ∈ ℝ)
14		peano2re 11148	. . . . . . 7 ⊢ ((abs‘𝑎) ∈ ℝ → ((abs‘𝑎) + 1) ∈ ℝ)
15	10, 14	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ)
16	15	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) ∈ ℝ)
17	13, 16	ifclda 4500	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ∈ ℝ)
18	1, 17	eqeltrid 2845	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
19		0re 10978	. . . . 5 ⊢ 0 ∈ ℝ
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ)
21	9	absge0d 15154	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎))
22		breq2 5083	. . . . . 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 5083	. . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
25	24, 2	eleqtrdi 2851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (^◡abs “ (0[,)𝑅)))
26		ffn 6598	. . . . . . . . . . . . 13 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27		elpreima 6932	. . . . . . . . . . . . 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 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
30	29	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
31		iccssxr 13161	. . . . . . . . . . . 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 25574	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
36	35	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
37	31, 36	sselid 3924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ^*)
38		elico2 13142	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
41	40	simp3d 1143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
42	41	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
43		avglt1 12211	. . . . . . . 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 231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
46	10	ltp1d 11905	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
47	46	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
48	22, 23, 45, 47	ifbothda 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
49	48, 1	breqtrrdi 5121	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
50	20, 10, 18, 21, 49	lelttrd 11133	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < 𝑀)
51	18, 50	elrpd 12768	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ⁺)
52		breq1 5082	. . . 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 5082	. . . 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 12212	. . . . . 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 231	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) < 𝑅)
57	15	rexrd 11026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ^*)
58	37, 57	xrlenltd 11042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) ↔ ¬ ((abs‘𝑎) + 1) < 𝑅))
59		0xr 11023	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
60		pnfxr 11030	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
61		elicc1 13122	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^ ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)))
62	59, 60, 61	mp2an 689	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
63	35, 62	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
64	63	simp2d 1142	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑅)
65	64	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ 𝑅)
66		ge0gtmnf 12905	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → -∞ < 𝑅)
67	37, 65, 66	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → -∞ < 𝑅)
68		xrre 12902	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ (-∞ < 𝑅 ∧ 𝑅 ≤ ((abs‘𝑎) + 1))) → 𝑅 ∈ ℝ)
69	68	expr 457	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ -∞ < 𝑅) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
70	37, 15, 67, 69	syl21anc 835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
71	58, 70	sylbird 259	. . . . . 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 407	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) < 𝑅)
74	52, 53, 56, 73	ifbothda 4503	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅)
75	1, 74	eqbrtrid 5114	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
76	51, 49, 75	3jca 1127	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ⁺ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 ifcif 4465 class class class wbr 5079 ↦ cmpt 5162 ^◡ccnv 5589 dom cdm 5590 “ cima 5593 Fn wfn 6427 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 supcsup 9177 ℂcc 10870 ℝcr 10871 0cc0 10872 1c1 10873 + caddc 10875 · cmul 10877 +∞cpnf 11007 -∞cmnf 11008 ℝ^*cxr 11009 < clt 11010 ≤ cle 11011 / cdiv 11632 2c2 12028 ℕ₀cn0 12233 ℝ⁺crp 12729 [,)cico 13080 [,]cicc 13081 seqcseq 13719 ↑cexp 13780 abscabs 14943 ⇝ cli 15191 Σcsu 15395

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-ico 13084 df-icc 13085 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195

This theorem is referenced by: psercn 25583 pserdvlem1 25584 pserdvlem2 25585 pserdv 25586

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