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Theorem psercnlem1 26809

Description: Lemma for psercn 26810. (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 6217	. . . . . . . . . . . 12 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
4		absf 15548	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
5	4	fdmi 6751	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
6	3, 5	sseqtri 3961	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
7	2, 6	eqsstri 3959	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	sselda 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
10	9	abscld 15650	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
11		readdcl 11322	. . . . . . 7 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 𝑅) ∈ ℝ)
12	10, 11	sylan 592	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 𝑅) ∈ ℝ)
13	12	rehalfcld 12633	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) ∈ ℝ)
14		peano2re 11522	. . . . . . 7 ⊢ ((abs‘𝑎) ∈ ℝ → ((abs‘𝑎) + 1) ∈ ℝ)
15	10, 14	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ)
16	15	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) ∈ ℝ)
17	13, 16	ifclda 4521	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ∈ ℝ)
18	1, 17	eqeltrid 2868	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
19		0re 11349	. . . . 5 ⊢ 0 ∈ ℝ
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ)
21	9	absge0d 15658	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎))
22		breq2 5112	. . . . . 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 5112	. . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
25	24, 2	eleqtrdi 2874	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (^◡abs “ (0[,)𝑅)))
26		ffn 6739	. . . . . . . . . . . . 13 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27		elpreima 7088	. . . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
30	29	simprd 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
31		iccssxr 13601	. . . . . . . . . . . 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 26801	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
36	35	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
37	31, 36	sselid 3911	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ^*)
38		elico2 13581	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
39	19, 37, 38	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
40	30, 39	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
41	40	simp3d 1162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑅)
42	41	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
43		avglt1 12624	. . . . . . . 8 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
44	10, 43	sylan 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
45	42, 44	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
46	10	ltp1d 12287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
47	46	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
48	22, 23, 45, 47	ifbothda 4524	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
49	48, 1	breqtrrdi 5153	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
50	20, 10, 18, 21, 49	lelttrd 11507	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < 𝑀)
51	18, 50	elrpd 13201	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ⁺)
52		breq1 5111	. . . 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 5111	. . . 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 12625	. . . . . 6 ⊢ (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (((abs‘𝑎) + 𝑅) / 2) < 𝑅))
55	10, 54	sylan 592	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (((abs‘𝑎) + 𝑅) / 2) < 𝑅))
56	42, 55	mpbid 235	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) < 𝑅)
57	15	rexrd 11398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ^*)
58	37, 57	xrlenltd 11414	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) ↔ ¬ ((abs‘𝑎) + 1) < 𝑅))
59		0xr 11395	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
60		pnfxr 11402	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
61		elicc1 13560	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^ ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)))
62	59, 60, 61	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
63	35, 62	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))
64	63	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑅)
65	64	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ 𝑅)
66		ge0gtmnf 13342	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → -∞ < 𝑅)
67	37, 65, 66	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → -∞ < 𝑅)
68		xrre 13339	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ (-∞ < 𝑅 ∧ 𝑅 ≤ ((abs‘𝑎) + 1))) → 𝑅 ∈ ℝ)
69	68	expr 462	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ^* ∧ ((abs‘𝑎) + 1) ∈ ℝ) ∧ -∞ < 𝑅) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
70	37, 15, 67, 69	syl21anc 851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) → 𝑅 ∈ ℝ))
71	58, 70	sylbird 263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (¬ ((abs‘𝑎) + 1) < 𝑅 → 𝑅 ∈ ℝ))
72	71	con1d 146	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (¬ 𝑅 ∈ ℝ → ((abs‘𝑎) + 1) < 𝑅))
73	72	imp 412	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ ¬ 𝑅 ∈ ℝ) → ((abs‘𝑎) + 1) < 𝑅)
74	52, 53, 56, 73	ifbothda 4524	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅)
75	1, 74	eqbrtrid 5146	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 < 𝑅)
76	51, 49, 75	3jca 1146	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑀 ∈ ℝ⁺ ∧ (abs‘𝑎) < 𝑀 ∧ 𝑀 < 𝑅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {crab 3416 ⊆ wss 3900 ifcif 4485 class class class wbr 5108 ↦ cmpt 5192 ^◡ccnv 5657 dom cdm 5658 “ cima 5661 Fn wfn 6561 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 supcsup 9469 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 / cdiv 12013 2c2 12437 ℕ₀cn0 12646 ℝ⁺crp 13160 [,)cico 13518 [,]cicc 13519 seqcseq 14184 ↑cexp 14244 abscabs 15444 ⇝ cli 15695 Σcsu 15897

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-ico 13522 df-icc 13523 df-fz 13680 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699

This theorem is used by: psercn 26810 pserdvlem1 26811 pserdvlem2 26812 pserdv 26813

W3C validator