pntlemc - Metamath Proof Explorer

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Theorem pntlemc 25084

Description: Lemma for pnt 25103. 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 25083	. . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp2d 1067	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
11	2, 10	rpdivcld 11765	. . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ⁺)
12	1, 11	syl5eqel 2692	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
14	5, 12	rpdivcld 11765	. . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
15	14	rpred 11748	. . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
16	15	rpefcld 14674	. . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ⁺)
17	13, 16	syl5eqel 2692	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
18	12	rpred 11748	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
19	12	rpgt0d 11751	. . . 4 ⊢ (𝜑 → 0 < 𝐸)
20	2	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ)
21	4	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
22	10	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23		pntlem1.u2	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
24	21	ltp1d 10833	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
25	24, 7	syl6breqr 4625	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷)
26	20, 21, 22, 23, 25	lelttrd 10074	. . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷)
27	10	rpcnd 11750	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	27	mulid1d 9936	. . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷)
29	26, 28	breqtrrd 4611	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1))
30		1red 9934	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
31	20, 30, 10	ltdivmuld 11799	. . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1)))
32	29, 31	mpbird 246	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1)
33	1, 32	syl5eqbr 4618	. . . 4 ⊢ (𝜑 → 𝐸 < 1)
34		0xr 9965	. . . . 5 ⊢ 0 ∈ ℝ^*
35		1re 9918	. . . . . 6 ⊢ 1 ∈ ℝ
36	35	rexri 9976	. . . . 5 ⊢ 1 ∈ ℝ^*
37		elioo2 12087	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)))
38	34, 36, 37	mp2an 704	. . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))
39	18, 19, 33, 38	syl3anbrc 1239	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
40		efgt1 14685	. . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ⁺ → 1 < (exp‘(𝐵 / 𝐸)))
41	14, 40	syl 17	. . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸)))
42	41, 13	syl6breqr 4625	. . 3 ⊢ (𝜑 → 1 < 𝐾)
43		ltaddrp 11743	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → 1 < (1 + 𝐴))
44	35, 4, 43	sylancr 694	. . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴))
45	2	rpcnne0d 11757	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0))
46		divid 10593	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1)
48	4	rpcnd 11750	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49		ax-1cn 9873	. . . . . . . . 9 ⊢ 1 ∈ ℂ
50		addcom 10101	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴))
51	48, 49, 50	sylancl 693	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴))
52	7, 51	syl5eq 2656	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴))
53	44, 47, 52	3brtr4d 4615	. . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷)
54	20, 2, 10, 53	ltdiv23d 11813	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈)
55	1, 54	syl5eqbr 4618	. . . 4 ⊢ (𝜑 → 𝐸 < 𝑈)
56		difrp 11744	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
57	18, 20, 56	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
58	55, 57	mpbid 221	. . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
59	39, 42, 58	3jca 1235	. 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
60	12, 17, 59	3jca 1235	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 3c3 10948 cdc 11369 ℝ⁺crp 11708 (,)cioo 12046 ↑cexp 12722 expce 14631 ψcchp 24619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637

This theorem is referenced by: pntlema 25085 pntlemb 25086 pntlemg 25087 pntlemh 25088 pntlemq 25090 pntlemr 25091 pntlemj 25092 pntlemi 25093 pntlemf 25094 pntlemo 25096 pntleme 25097 pntlemp 25099

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