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Theorem pntlema 26725

Description: Lemma for pnt 26743. Closure for the constants used in the proof. The mammoth expression 𝑊 is a number large enough to satisfy all the lower bounds needed for 𝑍. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑌 is x₂, 𝑋 is x₁, 𝐶 is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and 𝑊 is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
pntlem1.e	⊢ 𝐸 = (𝑈 / 𝐷)
pntlem1.k	⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y	⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
pntlem1.x	⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
pntlem1.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))

**Assertion**
Ref	Expression
pntlema	⊢ (𝜑 → 𝑊 ∈ ℝ⁺)

Distinct variable group: 𝐸,𝑎 Allowed substitution hints: 𝜑(𝑎) 𝐴(𝑎) 𝐵(𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑅(𝑎) 𝑈(𝑎) 𝐹(𝑎) 𝐾(𝑎) 𝐿(𝑎) 𝑊(𝑎) 𝑋(𝑎) 𝑌(𝑎)

**Proof of Theorem pntlema**
Step	Hyp	Ref	Expression
1		pntlem1.w	. 2 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
2		pntlem1.y	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
3	2	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
4		4nn 12039	. . . . . . 7 ⊢ 4 ∈ ℕ
5		nnrp 12723	. . . . . . 7 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ 4 ∈ ℝ⁺
7		pntlem1.r	. . . . . . . . 9 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
8		pntlem1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
9		pntlem1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
10		pntlem1.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
11		pntlem1.d	. . . . . . . . 9 ⊢ 𝐷 = (𝐴 + 1)
12		pntlem1.f	. . . . . . . . 9 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
13	7, 8, 9, 10, 11, 12	pntlemd 26723	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
14	13	simp1d 1140	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
15		pntlem1.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
16		pntlem1.u2	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
17		pntlem1.e	. . . . . . . . 9 ⊢ 𝐸 = (𝑈 / 𝐷)
18		pntlem1.k	. . . . . . . . 9 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
19	7, 8, 9, 10, 11, 12, 15, 16, 17, 18	pntlemc 26724	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
20	19	simp1d 1140	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
21	14, 20	rpmulcld 12770	. . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
22		rpdivcl 12737	. . . . . 6 ⊢ ((4 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (4 / (𝐿 · 𝐸)) ∈ ℝ⁺)
23	6, 21, 22	sylancr 586	. . . . 5 ⊢ (𝜑 → (4 / (𝐿 · 𝐸)) ∈ ℝ⁺)
24	3, 23	rpaddcld 12769	. . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ⁺)
25		2z 12335	. . . 4 ⊢ 2 ∈ ℤ
26		rpexpcl 13782	. . . 4 ⊢ (((𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ⁺)
27	24, 25, 26	sylancl 585	. . 3 ⊢ (𝜑 → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ⁺)
28		pntlem1.x	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
29	28	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
30	19	simp2d 1141	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
31		rpexpcl 13782	. . . . . . 7 ⊢ ((𝐾 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ⁺)
32	30, 25, 31	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ⁺)
33	29, 32	rpmulcld 12770	. . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ⁺)
34		4z 12337	. . . . 5 ⊢ 4 ∈ ℤ
35		rpexpcl 13782	. . . . 5 ⊢ (((𝑋 · (𝐾↑2)) ∈ ℝ⁺ ∧ 4 ∈ ℤ) → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ⁺)
36	33, 34, 35	sylancl 585	. . . 4 ⊢ (𝜑 → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ⁺)
37		3nn0 12234	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
38		2nn 12029	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
39	37, 38	decnncl 12439	. . . . . . . . . 10 ⊢ ;32 ∈ ℕ
40		nnrp 12723	. . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ⁺)
41	39, 40	ax-mp 5	. . . . . . . . 9 ⊢ ;32 ∈ ℝ⁺
42		rpmulcl 12735	. . . . . . . . 9 ⊢ ((;32 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (;32 · 𝐵) ∈ ℝ⁺)
43	41, 9, 42	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ⁺)
44	19	simp3d 1142	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
45	44	simp3d 1142	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
46		rpexpcl 13782	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ⁺)
47	20, 25, 46	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ⁺)
48	14, 47	rpmulcld 12770	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ⁺)
49	45, 48	rpmulcld 12770	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ⁺)
50	43, 49	rpdivcld 12771	. . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ⁺)
51		3rp 12718	. . . . . . . . 9 ⊢ 3 ∈ ℝ⁺
52		rpmulcl 12735	. . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ⁺ ∧ 3 ∈ ℝ⁺) → (𝑈 · 3) ∈ ℝ⁺)
53	15, 51, 52	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ⁺)
54		pntlem1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
55	53, 54	rpaddcld 12769	. . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ⁺)
56	50, 55	rpmulcld 12770	. . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ⁺)
57	56	rpred 12754	. . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ)
58	57	rpefcld 15795	. . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ⁺)
59	36, 58	rpaddcld 12769	. . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ⁺)
60	27, 59	rpaddcld 12769	. 2 ⊢ (𝜑 → (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) ∈ ℝ⁺)
61	1, 60	eqeltrid 2844	1 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 class class class wbr 5078 ↦ cmpt 5161 ‘cfv 6430 (class class class)co 7268 0cc0 10855 1c1 10856 + caddc 10858 · cmul 10860 < clt 10993 ≤ cle 10994 − cmin 11188 / cdiv 11615 ℕcn 11956 2c2 12011 3c3 12012 4c4 12013 ℤcz 12302 cdc 12419 ℝ⁺crp 12712 (,)cioo 13061 ↑cexp 13763 expce 15752 ψcchp 26223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-rp 12713 df-ioo 13065 df-ico 13067 df-fz 13222 df-fzo 13365 df-fl 13493 df-seq 13703 df-exp 13764 df-fac 13969 df-bc 13998 df-hash 14026 df-shft 14759 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-limsup 15161 df-clim 15178 df-rlim 15179 df-sum 15379 df-ef 15758

This theorem is referenced by: pntlemb 26726 pntleme 26737

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