Theorem log2ublem1 26923

Description: Lemma for log2ub 26926. The proof of log2ub 26926, which is simply the evaluation of log2tlbnd 26922 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 12251	. . . . . . . 8 ⊢ 3 ∈ ℕ
3		7nn0 12450	. . . . . . . 8 ⊢ 7 ∈ ℕ0
4		nnexpcl 14027	. . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
5	2, 3, 4	mp2an 693	. . . . . . 7 ⊢ (3↑7) ∈ ℕ
6		5nn 12258	. . . . . . . 8 ⊢ 5 ∈ ℕ
7		7nn 12264	. . . . . . . 8 ⊢ 7 ∈ ℕ
8	6, 7	nnmulcli 12190	. . . . . . 7 ⊢ (5 · 7) ∈ ℕ
9	5, 8	nnmulcli 12190	. . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ
10	9	nncni 12175	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ
11		log2ublem1.3	. . . . . 6 ⊢ 𝐷 ∈ ℕ0
12	11	nn0cni 12440	. . . . 5 ⊢ 𝐷 ∈ ℂ
13		log2ublem1.4	. . . . . 6 ⊢ 𝐸 ∈ ℕ
14	13	nncni 12175	. . . . 5 ⊢ 𝐸 ∈ ℂ
15	13	nnne0i 12208	. . . . 5 ⊢ 𝐸 ≠ 0
16	10, 12, 14, 15	divassi 11902	. . . 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 12446	. . . . . . . . . 10 ⊢ 3 ∈ ℕ0
19	18, 3	nn0expcli 14041	. . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0
20		5nn0 12448	. . . . . . . . . 10 ⊢ 5 ∈ ℕ0
21	20, 3	nn0mulcli 12466	. . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0
22	19, 21	nn0mulcli 12466	. . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0
23	22, 11	nn0mulcli 12466	. . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
24	23	nn0rei 12439	. . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25		log2ublem1.6	. . . . . . 7 ⊢ 𝐹 ∈ ℕ0
26	25	nn0rei 12439	. . . . . 6 ⊢ 𝐹 ∈ ℝ
27	13	nnrei 12174	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
28	13	nngt0i 12207	. . . . . . 7 ⊢ 0 < 𝐸
29	27, 28	pm3.2i 470	. . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30		ledivmul 12023	. . . . . 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 1464	. . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
32	17, 31	mpbir 231	. . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
33	16, 32	eqbrtrri 5109	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
34	9	nnrei 12174	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ
35		log2ublem1.2	. . . . 5 ⊢ 𝐴 ∈ ℝ
36	34, 35	remulcli 11152	. . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
37	11	nn0rei 12439	. . . . . 6 ⊢ 𝐷 ∈ ℝ
38		nndivre 12209	. . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
39	37, 13, 38	mp2an 693	. . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ
40	34, 39	remulcli 11152	. . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41		log2ublem1.5	. . . . 5 ⊢ 𝐵 ∈ ℕ0
42	41	nn0rei 12439	. . . 4 ⊢ 𝐵 ∈ ℝ
43	36, 40, 42, 26	le2addi 11704	. . 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 693	. 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 7371	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
47	35	recni 11150	. . . 4 ⊢ 𝐴 ∈ ℂ
48	39	recni 11150	. . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ
49	10, 47, 48	adddii 11148	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
50	46, 49	eqtr2i 2761	. 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 5115	1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  (class class class)co 7360  ℝcr 11028  0cc0 11029   + caddc 11032   · cmul 11034   < clt 11170   ≤ cle 11171   / cdiv 11798  ℕcn 12165  3c3 12228  5c5 12230  7c7 12232  ℕ₀cn0 12428  ↑cexp 14014

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 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955  df-exp 14015

This theorem is referenced by:  log2ublem2  26924  log2ub  26926