log2ublem1 - Metamath Proof Explorer

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Theorem log2ublem1 26871

Description: Lemma for log2ub 26874. The proof of log2ub 26874, which is simply the evaluation of log2tlbnd 26870 for 𝑁 = 4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator 𝑑 (usually a large power of 10) and work with the closest approximations of the form 𝑛 / 𝑑 for some integer 𝑛 instead. It turns out that for our purposes it is sufficient to take 𝑑 = (3↑7) · 5 · 7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)

**Hypotheses**
Ref	Expression
log2ublem1.1	⊢ (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵
log2ublem1.2	⊢ 𝐴 ∈ ℝ
log2ublem1.3	⊢ 𝐷 ∈ ℕ₀
log2ublem1.4	⊢ 𝐸 ∈ ℕ
log2ublem1.5	⊢ 𝐵 ∈ ℕ₀
log2ublem1.6	⊢ 𝐹 ∈ ℕ₀
log2ublem1.7	⊢ 𝐶 = (𝐴 + (𝐷 / 𝐸))
log2ublem1.8	⊢ (𝐵 + 𝐹) = 𝐺
log2ublem1.9	⊢ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)

**Assertion**
Ref	Expression
log2ublem1	⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

**Proof of Theorem log2ublem1**
Step	Hyp	Ref	Expression
1		log2ublem1.1	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵
2		3nn 12315	. . . . . . . 8 ⊢ 3 ∈ ℕ
3		7nn0 12518	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
4		nnexpcl 14065	. . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ₀) → (3↑7) ∈ ℕ)
5	2, 3, 4	mp2an 691	. . . . . . 7 ⊢ (3↑7) ∈ ℕ
6		5nn 12322	. . . . . . . 8 ⊢ 5 ∈ ℕ
7		7nn 12328	. . . . . . . 8 ⊢ 7 ∈ ℕ
8	6, 7	nnmulcli 12261	. . . . . . 7 ⊢ (5 · 7) ∈ ℕ
9	5, 8	nnmulcli 12261	. . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ
10	9	nncni 12246	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ
11		log2ublem1.3	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12	11	nn0cni 12508	. . . . 5 ⊢ 𝐷 ∈ ℂ
13		log2ublem1.4	. . . . . 6 ⊢ 𝐸 ∈ ℕ
14	13	nncni 12246	. . . . 5 ⊢ 𝐸 ∈ ℂ
15	13	nnne0i 12276	. . . . 5 ⊢ 𝐸 ≠ 0
16	10, 12, 14, 15	divassi 11994	. . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) = (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))
17		log2ublem1.9	. . . . 5 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)
18		3nn0 12514	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
19	18, 3	nn0expcli 14079	. . . . . . . . 9 ⊢ (3↑7) ∈ ℕ₀
20		5nn0 12516	. . . . . . . . . 10 ⊢ 5 ∈ ℕ₀
21	20, 3	nn0mulcli 12534	. . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ₀
22	19, 21	nn0mulcli 12534	. . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ₀
23	22, 11	nn0mulcli 12534	. . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ₀
24	23	nn0rei 12507	. . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25		log2ublem1.6	. . . . . . 7 ⊢ 𝐹 ∈ ℕ₀
26	25	nn0rei 12507	. . . . . 6 ⊢ 𝐹 ∈ ℝ
27	13	nnrei 12245	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
28	13	nngt0i 12275	. . . . . . 7 ⊢ 0 < 𝐸
29	27, 28	pm3.2i 470	. . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30		ledivmul 12114	. . . . . 6 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ ∧ 𝐹 ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)))
31	24, 26, 29, 30	mp3an 1458	. . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
32	17, 31	mpbir 230	. . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
33	16, 32	eqbrtrri 5165	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
34	9	nnrei 12245	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ
35		log2ublem1.2	. . . . 5 ⊢ 𝐴 ∈ ℝ
36	34, 35	remulcli 11254	. . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
37	11	nn0rei 12507	. . . . . 6 ⊢ 𝐷 ∈ ℝ
38		nndivre 12277	. . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
39	37, 13, 38	mp2an 691	. . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ
40	34, 39	remulcli 11254	. . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41		log2ublem1.5	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
42	41	nn0rei 12507	. . . 4 ⊢ 𝐵 ∈ ℝ
43	36, 40, 42, 26	le2addi 11801	. . 3 ⊢ (((((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵 ∧ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹) → ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹))
44	1, 33, 43	mp2an 691	. 2 ⊢ ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹)
45		log2ublem1.7	. . . 4 ⊢ 𝐶 = (𝐴 + (𝐷 / 𝐸))
46	45	oveq2i 7425	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
47	35	recni 11252	. . . 4 ⊢ 𝐴 ∈ ℂ
48	39	recni 11252	. . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ
49	10, 47, 48	adddii 11250	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
50	46, 49	eqtr2i 2757	. 2 ⊢ ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) = (((3↑7) · (5 · 7)) · 𝐶)
51		log2ublem1.8	. 2 ⊢ (𝐵 + 𝐹) = 𝐺
52	44, 50, 51	3brtr3i 5171	1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 0cc0 11132 + caddc 11135 · cmul 11137 < clt 11272 ≤ cle 11273 / cdiv 11895 ℕcn 12236 3c3 12292 5c5 12294 7c7 12296 ℕ₀cn0 12496 ↑cexp 14052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 df-exp 14053

This theorem is referenced by: log2ublem2 26872 log2ub 26874

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