pntlemc - Metamath Proof Explorer

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Theorem pntlemc 27558

Description: Lemma for pnt 27577. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑈 is α, 𝐸 is ε, and 𝐾 is K. (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))

**Assertion**
Ref	Expression
pntlemc	⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))

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

**Proof of Theorem pntlemc**
Step	Hyp	Ref	Expression
1		pntlem1.e	. . 3 ⊢ 𝐸 = (𝑈 / 𝐷)
2		pntlem1.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
3		pntlem1.r	. . . . . 6 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4		pntlem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
5		pntlem1.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
6		pntlem1.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
7		pntlem1.d	. . . . . 6 ⊢ 𝐷 = (𝐴 + 1)
8		pntlem1.f	. . . . . 6 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
9	3, 4, 5, 6, 7, 8	pntlemd 27557	. . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
11	2, 10	rpdivcld 13068	. . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ⁺)
12	1, 11	eqeltrid 2838	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
14	5, 12	rpdivcld 13068	. . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
15	14	rpred 13051	. . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
16	15	rpefcld 16123	. . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ⁺)
17	13, 16	eqeltrid 2838	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
18	12	rpred 13051	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
19	12	rpgt0d 13054	. . . 4 ⊢ (𝜑 → 0 < 𝐸)
20	2	rpred 13051	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ)
21	4	rpred 13051	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
22	10	rpred 13051	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23		pntlem1.u2	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
24	21	ltp1d 12172	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
25	24, 7	breqtrrdi 5161	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷)
26	20, 21, 22, 23, 25	lelttrd 11393	. . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷)
27	10	rpcnd 13053	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	27	mulridd 11252	. . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷)
29	26, 28	breqtrrd 5147	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1))
30		1red 11236	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
31	20, 30, 10	ltdivmuld 13102	. . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1)))
32	29, 31	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1)
33	1, 32	eqbrtrid 5154	. . . 4 ⊢ (𝜑 → 𝐸 < 1)
34		0xr 11282	. . . . 5 ⊢ 0 ∈ ℝ^*
35		1xr 11294	. . . . 5 ⊢ 1 ∈ ℝ^*
36		elioo2 13403	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)))
37	34, 35, 36	mp2an 692	. . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))
38	18, 19, 33, 37	syl3anbrc 1344	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
39		efgt1 16134	. . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ⁺ → 1 < (exp‘(𝐵 / 𝐸)))
40	14, 39	syl 17	. . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸)))
41	40, 13	breqtrrdi 5161	. . 3 ⊢ (𝜑 → 1 < 𝐾)
42		1re 11235	. . . . . . . 8 ⊢ 1 ∈ ℝ
43		ltaddrp 13046	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → 1 < (1 + 𝐴))
44	42, 4, 43	sylancr 587	. . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴))
45	2	rpcnne0d 13060	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0))
46		divid 11927	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1)
48	4	rpcnd 13053	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49		ax-1cn 11187	. . . . . . . . 9 ⊢ 1 ∈ ℂ
50		addcom 11421	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴))
51	48, 49, 50	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴))
52	7, 51	eqtrid 2782	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴))
53	44, 47, 52	3brtr4d 5151	. . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷)
54	20, 2, 10, 53	ltdiv23d 13118	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈)
55	1, 54	eqbrtrid 5154	. . . 4 ⊢ (𝜑 → 𝐸 < 𝑈)
56		difrp 13047	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
57	18, 20, 56	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
58	55, 57	mpbid 232	. . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
59	38, 41, 58	3jca 1128	. 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
60	12, 17, 59	3jca 1128	1 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ↦ cmpt 5201 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 2c2 12295 3c3 12296 cdc 12708 ℝ⁺crp 13008 (,)cioo 13362 ↑cexp 14079 expce 16077 ψcchp 27055

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-ioo 13366 df-ico 13368 df-fz 13525 df-fzo 13672 df-fl 13809 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083

This theorem is referenced by: pntlema 27559 pntlemb 27560 pntlemg 27561 pntlemh 27562 pntlemq 27564 pntlemr 27565 pntlemj 27566 pntlemi 27567 pntlemf 27568 pntlemo 27570 pntleme 27571 pntlemp 27573

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