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Theorem psercnlem1 26619

Description: Lemma for psercn 26620. (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 6086	. . . . . . . . . . . 12 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
4		absf 15409	. . . . . . . . . . . . 13 ⊢ abs:ℂ⟶ℝ
5	4	fdmi 6721	. . . . . . . . . . . 12 ⊢ dom abs = ℂ
6	3, 5	sseqtri 3986	. . . . . . . . . . 11 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ ℂ
7	2, 6	eqsstri 3984	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℂ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
9	8	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ℂ)
10	9	abscld 15510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) ∈ ℝ)
11		readdcl 11194	. . . . . . 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 12502	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑅 ∈ ℝ) → (((abs‘𝑎) + 𝑅) / 2) ∈ ℝ)
14		peano2re 11394	. . . . . . 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 4525	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) ∈ ℝ)
18	1, 17	eqeltrid 2869	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ)
19		0re 11221	. . . . 5 ⊢ 0 ∈ ℝ
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ∈ ℝ)
21	9	absge0d 15518	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 ≤ (abs‘𝑎))
22		breq2 5115	. . . . . 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 5115	. . . . . 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 2875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (^◡abs “ (0[,)𝑅)))
26		ffn 6709	. . . . . . . . . . . . 13 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27		elpreima 7057	. . . . . . . . . . . . 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 13469	. . . . . . . . . . . 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 26611	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞))
36	35	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ (0[,]+∞))
37	31, 36	sselid 3936	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑅 ∈ ℝ^*)
38		elico2 13449	. . . . . . . . . . 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 12493	. . . . . . . 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 12156	. . . . . . 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 4528	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
49	48, 1	breqtrrdi 5155	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (abs‘𝑎) < 𝑀)
50	20, 10, 18, 21, 49	lelttrd 11379	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 0 < 𝑀)
51	18, 50	elrpd 13069	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑀 ∈ ℝ⁺)
52		breq1 5114	. . . 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 5114	. . . 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 12494	. . . . . 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 11270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((abs‘𝑎) + 1) ∈ ℝ^*)
58	37, 57	xrlenltd 11286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑅 ≤ ((abs‘𝑎) + 1) ↔ ¬ ((abs‘𝑎) + 1) < 𝑅))
59		0xr 11267	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
60		pnfxr 11274	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
61		elicc1 13428	. . . . . . . . . . . . 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 13210	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → -∞ < 𝑅)
67	37, 65, 66	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → -∞ < 𝑅)
68		xrre 13207	. . . . . . . . 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 4528	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) < 𝑅)
75	1, 74	eqbrtrid 5148	. 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 2146 {crab 3418 ⊆ wss 3906 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 ^◡ccnv 5662 dom cdm 5663 “ cima 5666 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 supcsup 9403 ℂcc 11109 ℝcr 11110 0cc0 11111 1c1 11112 + caddc 11114 · cmul 11116 +∞cpnf 11251 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 / cdiv 11882 2c2 12306 ℕ₀cn0 12515 ℝ⁺crp 13028 [,)cico 13386 [,]cicc 13387 seqcseq 14051 ↑cexp 14111 abscabs 15305 ⇝ cli 15555 Σcsu 15757

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9405 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-n0 12516 df-z 12603 df-uz 12875 df-rp 13029 df-ico 13390 df-icc 13391 df-fz 13548 df-seq 14052 df-exp 14112 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-clim 15559

This theorem is used by: psercn 26620 pserdvlem1 26621 pserdvlem2 26622 pserdv 26623

W3C validator