pntlemc - Metamath Proof Explorer

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Theorem pntlemc 27927

Description: Lemma for pnt 27946. 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 27926	. . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp2d 1161	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
11	2, 10	rpdivcld 13184	. . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ⁺)
12	1, 11	eqeltrid 2864	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
14	5, 12	rpdivcld 13184	. . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
15	14	rpred 13167	. . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
16	15	rpefcld 16278	. . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ⁺)
17	13, 16	eqeltrid 2864	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
18	12	rpred 13167	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
19	12	rpgt0d 13170	. . . 4 ⊢ (𝜑 → 0 < 𝐸)
20	2	rpred 13167	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ)
21	4	rpred 13167	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
22	10	rpred 13167	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23		pntlem1.u2	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
24	21	ltp1d 12250	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
25	24, 7	breqtrrdi 5146	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷)
26	20, 21, 22, 23, 25	lelttrd 11471	. . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷)
27	10	rpcnd 13169	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	27	mulridd 11329	. . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷)
29	26, 28	breqtrrd 5132	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1))
30		1red 11312	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
31	20, 30, 10	ltdivmuld 13218	. . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1)))
32	29, 31	mpbird 260	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1)
33	1, 32	eqbrtrid 5139	. . . 4 ⊢ (𝜑 → 𝐸 < 1)
34		0xr 11359	. . . . 5 ⊢ 0 ∈ ℝ^*
35		1xr 11371	. . . . 5 ⊢ 1 ∈ ℝ^*
36		elioo2 13520	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)))
37	34, 35, 36	mp2an 705	. . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))
38	18, 19, 33, 37	syl3anbrc 1362	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
39		efgt1 16289	. . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ⁺ → 1 < (exp‘(𝐵 / 𝐸)))
40	14, 39	syl 18	. . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸)))
41	40, 13	breqtrrdi 5146	. . 3 ⊢ (𝜑 → 1 < 𝐾)
42		1re 11311	. . . . . . . 8 ⊢ 1 ∈ ℝ
43		ltaddrp 13162	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → 1 < (1 + 𝐴))
44	42, 4, 43	sylancr 599	. . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴))
45	2	rpcnne0d 13176	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0))
46		divid 12007	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1)
47	45, 46	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1)
48	4	rpcnd 13169	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49		ax-1cn 11261	. . . . . . . . 9 ⊢ 1 ∈ ℂ
50		addcom 11499	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴))
51	48, 49, 50	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴))
52	7, 51	eqtrid 2807	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴))
53	44, 47, 52	3brtr4d 5136	. . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷)
54	20, 2, 10, 53	ltdiv23d 13234	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈)
55	1, 54	eqbrtrid 5139	. . . 4 ⊢ (𝜑 → 𝐸 < 𝑈)
56		difrp 13163	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
57	18, 20, 56	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
58	55, 57	mpbid 235	. . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
59	38, 41, 58	3jca 1146	. 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
60	12, 17, 59	3jca 1146	1 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 − cmin 11544 / cdiv 11976 2c2 12400 3c3 12401 cdc 12817 ℝ⁺crp 13123 (,)cioo 13479 ↑cexp 14207 expce 16232 ψcchp 27425

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-er 8720 df-pm 8853 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-rp 13124 df-ioo 13483 df-ico 13485 df-fz 13643 df-fzo 13792 df-fl 13935 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-ef 16238

This theorem is used by: pntlema 27928 pntlemb 27929 pntlemg 27930 pntlemh 27931 pntlemq 27933 pntlemr 27934 pntlemj 27935 pntlemi 27936 pntlemf 27937 pntlemo 27939 pntleme 27940 pntlemp 27942

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