Theorem log2ublem1 26833

Description: Lemma for log2ub 26836. The proof of log2ub 26836, which is simply the evaluation of log2tlbnd 26832 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 12295	. . . . . . . 8 ⊢ 3 ∈ ℕ
3		7nn0 12498	. . . . . . . 8 ⊢ 7 ∈ ℕ0
4		nnexpcl 14045	. . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
5	2, 3, 4	mp2an 689	. . . . . . 7 ⊢ (3↑7) ∈ ℕ
6		5nn 12302	. . . . . . . 8 ⊢ 5 ∈ ℕ
7		7nn 12308	. . . . . . . 8 ⊢ 7 ∈ ℕ
8	6, 7	nnmulcli 12241	. . . . . . 7 ⊢ (5 · 7) ∈ ℕ
9	5, 8	nnmulcli 12241	. . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ
10	9	nncni 12226	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ
11		log2ublem1.3	. . . . . 6 ⊢ 𝐷 ∈ ℕ0
12	11	nn0cni 12488	. . . . 5 ⊢ 𝐷 ∈ ℂ
13		log2ublem1.4	. . . . . 6 ⊢ 𝐸 ∈ ℕ
14	13	nncni 12226	. . . . 5 ⊢ 𝐸 ∈ ℂ
15	13	nnne0i 12256	. . . . 5 ⊢ 𝐸 ≠ 0
16	10, 12, 14, 15	divassi 11974	. . . 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 12494	. . . . . . . . . 10 ⊢ 3 ∈ ℕ0
19	18, 3	nn0expcli 14059	. . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0
20		5nn0 12496	. . . . . . . . . 10 ⊢ 5 ∈ ℕ0
21	20, 3	nn0mulcli 12514	. . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0
22	19, 21	nn0mulcli 12514	. . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0
23	22, 11	nn0mulcli 12514	. . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
24	23	nn0rei 12487	. . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25		log2ublem1.6	. . . . . . 7 ⊢ 𝐹 ∈ ℕ0
26	25	nn0rei 12487	. . . . . 6 ⊢ 𝐹 ∈ ℝ
27	13	nnrei 12225	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
28	13	nngt0i 12255	. . . . . . 7 ⊢ 0 < 𝐸
29	27, 28	pm3.2i 470	. . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30		ledivmul 12094	. . . . . 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 1457	. . . . 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 5164	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
34	9	nnrei 12225	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ
35		log2ublem1.2	. . . . 5 ⊢ 𝐴 ∈ ℝ
36	34, 35	remulcli 11234	. . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
37	11	nn0rei 12487	. . . . . 6 ⊢ 𝐷 ∈ ℝ
38		nndivre 12257	. . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
39	37, 13, 38	mp2an 689	. . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ
40	34, 39	remulcli 11234	. . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41		log2ublem1.5	. . . . 5 ⊢ 𝐵 ∈ ℕ0
42	41	nn0rei 12487	. . . 4 ⊢ 𝐵 ∈ ℝ
43	36, 40, 42, 26	le2addi 11781	. . 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 689	. 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 7416	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
47	35	recni 11232	. . . 4 ⊢ 𝐴 ∈ ℂ
48	39	recni 11232	. . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ
49	10, 47, 48	adddii 11230	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
50	46, 49	eqtr2i 2755	. 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 5170	1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  (class class class)co 7405  ℝcr 11111  0cc0 11112   + caddc 11115   · cmul 11117   < clt 11252   ≤ cle 11253   / cdiv 11875  ℕcn 12216  3c3 12272  5c5 12274  7c7 12276  ℕ₀cn0 12476  ↑cexp 14032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13973  df-exp 14033

This theorem is referenced by:  log2ublem2  26834  log2ub  26836