pntlemc - Metamath Proof Explorer

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Theorem pntlemc 27583

Description: Lemma for pnt 27602. 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 27582	. . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp2d 1149	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
11	2, 10	rpdivcld 13001	. . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ⁺)
12	1, 11	eqeltrid 2844	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
14	5, 12	rpdivcld 13001	. . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
15	14	rpred 12984	. . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
16	15	rpefcld 16070	. . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ⁺)
17	13, 16	eqeltrid 2844	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
18	12	rpred 12984	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
19	12	rpgt0d 12987	. . . 4 ⊢ (𝜑 → 0 < 𝐸)
20	2	rpred 12984	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ)
21	4	rpred 12984	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
22	10	rpred 12984	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23		pntlem1.u2	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
24	21	ltp1d 12084	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
25	24, 7	breqtrrdi 5121	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷)
26	20, 21, 22, 23, 25	lelttrd 11302	. . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷)
27	10	rpcnd 12986	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	27	mulridd 11160	. . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷)
29	26, 28	breqtrrd 5107	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1))
30		1red 11143	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
31	20, 30, 10	ltdivmuld 13035	. . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1)))
32	29, 31	mpbird 258	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1)
33	1, 32	eqbrtrid 5114	. . . 4 ⊢ (𝜑 → 𝐸 < 1)
34		0xr 11190	. . . . 5 ⊢ 0 ∈ ℝ^*
35		1xr 11202	. . . . 5 ⊢ 1 ∈ ℝ^*
36		elioo2 13337	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)))
37	34, 35, 36	mp2an 698	. . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))
38	18, 19, 33, 37	syl3anbrc 1350	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
39		efgt1 16081	. . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ⁺ → 1 < (exp‘(𝐵 / 𝐸)))
40	14, 39	syl 17	. . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸)))
41	40, 13	breqtrrdi 5121	. . 3 ⊢ (𝜑 → 1 < 𝐾)
42		1re 11142	. . . . . . . 8 ⊢ 1 ∈ ℝ
43		ltaddrp 12979	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → 1 < (1 + 𝐴))
44	42, 4, 43	sylancr 593	. . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴))
45	2	rpcnne0d 12993	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0))
46		divid 11838	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1)
48	4	rpcnd 12986	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49		ax-1cn 11094	. . . . . . . . 9 ⊢ 1 ∈ ℂ
50		addcom 11330	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴))
51	48, 49, 50	sylancl 592	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴))
52	7, 51	eqtrid 2787	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴))
53	44, 47, 52	3brtr4d 5111	. . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷)
54	20, 2, 10, 53	ltdiv23d 13051	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈)
55	1, 54	eqbrtrid 5114	. . . 4 ⊢ (𝜑 → 𝐸 < 𝑈)
56		difrp 12980	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
57	18, 20, 56	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
58	55, 57	mpbid 233	. . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
59	38, 41, 58	3jca 1134	. 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
60	12, 17, 59	3jca 1134	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 class class class wbr 5079 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 / cdiv 11805 2c2 12234 3c3 12235 cdc 12642 ℝ⁺crp 12940 (,)cioo 13296 ↑cexp 14021 expce 16024 ψcchp 27081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-rp 12941 df-ioo 13300 df-ico 13302 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030

This theorem is referenced by: pntlema 27584 pntlemb 27585 pntlemg 27586 pntlemh 27587 pntlemq 27589 pntlemr 27590 pntlemj 27591 pntlemi 27592 pntlemf 27593 pntlemo 27595 pntleme 27596 pntlemp 27598

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