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 1144	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
11	2, 10	rpdivcld 13003	. . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ⁺)
12	1, 11	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
14	5, 12	rpdivcld 13003	. . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
15	14	rpred 12986	. . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
16	15	rpefcld 16072	. . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ⁺)
17	13, 16	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
18	12	rpred 12986	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
19	12	rpgt0d 12989	. . . 4 ⊢ (𝜑 → 0 < 𝐸)
20	2	rpred 12986	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ)
21	4	rpred 12986	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
22	10	rpred 12986	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23		pntlem1.u2	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
24	21	ltp1d 12086	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
25	24, 7	breqtrrdi 5127	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷)
26	20, 21, 22, 23, 25	lelttrd 11304	. . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷)
27	10	rpcnd 12988	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	27	mulridd 11162	. . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷)
29	26, 28	breqtrrd 5113	. . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1))
30		1red 11145	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
31	20, 30, 10	ltdivmuld 13037	. . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1)))
32	29, 31	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1)
33	1, 32	eqbrtrid 5120	. . . 4 ⊢ (𝜑 → 𝐸 < 1)
34		0xr 11192	. . . . 5 ⊢ 0 ∈ ℝ^*
35		1xr 11204	. . . . 5 ⊢ 1 ∈ ℝ^*
36		elioo2 13339	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)))
37	34, 35, 36	mp2an 693	. . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))
38	18, 19, 33, 37	syl3anbrc 1345	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
39		efgt1 16083	. . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ⁺ → 1 < (exp‘(𝐵 / 𝐸)))
40	14, 39	syl 17	. . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸)))
41	40, 13	breqtrrdi 5127	. . 3 ⊢ (𝜑 → 1 < 𝐾)
42		1re 11144	. . . . . . . 8 ⊢ 1 ∈ ℝ
43		ltaddrp 12981	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → 1 < (1 + 𝐴))
44	42, 4, 43	sylancr 588	. . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴))
45	2	rpcnne0d 12995	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0))
46		divid 11840	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1)
48	4	rpcnd 12988	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
49		ax-1cn 11096	. . . . . . . . 9 ⊢ 1 ∈ ℂ
50		addcom 11332	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴))
51	48, 49, 50	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴))
52	7, 51	eqtrid 2783	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴))
53	44, 47, 52	3brtr4d 5117	. . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷)
54	20, 2, 10, 53	ltdiv23d 13053	. . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈)
55	1, 54	eqbrtrid 5120	. . . 4 ⊢ (𝜑 → 𝐸 < 𝑈)
56		difrp 12982	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
57	18, 20, 56	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ⁺))
58	55, 57	mpbid 232	. . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
59	38, 41, 58	3jca 1129	. 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
60	12, 17, 59	3jca 1129	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 2c2 12236 3c3 12237 cdc 12644 ℝ⁺crp 12942 (,)cioo 13298 ↑cexp 14023 expce 16026 ψcchp 27056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-ioo 13302 df-ico 13304 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032

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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