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Theorem psercnlem1 26392

Description: Lemma for psercn 26393. (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 6074	. . . . . . . . . . . 12 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
4		absf 15361	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
5	4	fdmi 6722	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
6	3, 5	sseqtri 4012	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
7	2, 6	eqsstri 4010	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	sselda 3963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
10	9	abscld 15460	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
11		readdcl 11217	. . . . . . 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 12493	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) ∈ ℝ)
14		peano2re 11413	. . . . . . 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 4541	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ∈ ℝ)
18	1, 17	eqeltrid 2839	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
19		0re 11242	. . . . 5 ⊢ 0 ∈ ℝ
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ)
21	9	absge0d 15468	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎))
22		breq2 5128	. . . . . 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 5128	. . . . . 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 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (^◡abs “ (0[,)𝑅)))
26		ffn 6711	. . . . . . . . . . . . 13 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27		elpreima 7053	. . . . . . . . . . . . 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 13452	. . . . . . . . . . . 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 26383	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
37	31, 36	sselid 3961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ^*)
38		elico2 13432	. . . . . . . . . . 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 12484	. . . . . . . 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 12177	. . . . . . 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 4544	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
49	48, 1	breqtrrdi 5166	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
50	20, 10, 18, 21, 49	lelttrd 11398	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < 𝑀)
51	18, 50	elrpd 13053	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ⁺)
52		breq1 5127	. . . 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 5127	. . . 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 12485	. . . . . 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 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ^*)
58	37, 57	xrlenltd 11306	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) ↔ ¬ ((abs‘𝑎) + 1) < 𝑅))
59		0xr 11287	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
60		pnfxr 11294	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
61		elicc1 13411	. . . . . . . . . . . . 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 13193	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → -∞ < 𝑅)
67	37, 65, 66	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → -∞ < 𝑅)
68		xrre 13190	. . . . . . . . 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 4544	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅)
75	1, 74	eqbrtrid 5159	. 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 1540 ∈ wcel 2109 {crab 3420 ⊆ wss 3931 ifcif 4505 class class class wbr 5124 ↦ cmpt 5206 ^◡ccnv 5658 dom cdm 5659 “ cima 5662 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 supcsup 9457 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 +∞cpnf 11271 -∞cmnf 11272 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 / cdiv 11899 2c2 12300 ℕ₀cn0 12506 ℝ⁺crp 13013 [,)cico 13369 [,]cicc 13370 seqcseq 14024 ↑cexp 14084 abscabs 15258 ⇝ cli 15505 Σcsu 15707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-ico 13373 df-icc 13374 df-fz 13530 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509

This theorem is referenced by: psercn 26393 pserdvlem1 26394 pserdvlem2 26395 pserdv 26396

W3C validator