Theorem log2ublem1 27004

Description: Lemma for log2ub 27007. The proof of log2ub 27007, which is simply the evaluation of log2tlbnd 27003 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 12343	. . . . . . . 8 ⊢ 3 ∈ ℕ
3		7nn0 12546	. . . . . . . 8 ⊢ 7 ∈ ℕ0
4		nnexpcl 14112	. . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
5	2, 3, 4	mp2an 692	. . . . . . 7 ⊢ (3↑7) ∈ ℕ
6		5nn 12350	. . . . . . . 8 ⊢ 5 ∈ ℕ
7		7nn 12356	. . . . . . . 8 ⊢ 7 ∈ ℕ
8	6, 7	nnmulcli 12289	. . . . . . 7 ⊢ (5 · 7) ∈ ℕ
9	5, 8	nnmulcli 12289	. . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ
10	9	nncni 12274	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ
11		log2ublem1.3	. . . . . 6 ⊢ 𝐷 ∈ ℕ0
12	11	nn0cni 12536	. . . . 5 ⊢ 𝐷 ∈ ℂ
13		log2ublem1.4	. . . . . 6 ⊢ 𝐸 ∈ ℕ
14	13	nncni 12274	. . . . 5 ⊢ 𝐸 ∈ ℂ
15	13	nnne0i 12304	. . . . 5 ⊢ 𝐸 ≠ 0
16	10, 12, 14, 15	divassi 12021	. . . 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 12542	. . . . . . . . . 10 ⊢ 3 ∈ ℕ0
19	18, 3	nn0expcli 14126	. . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0
20		5nn0 12544	. . . . . . . . . 10 ⊢ 5 ∈ ℕ0
21	20, 3	nn0mulcli 12562	. . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0
22	19, 21	nn0mulcli 12562	. . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0
23	22, 11	nn0mulcli 12562	. . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
24	23	nn0rei 12535	. . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25		log2ublem1.6	. . . . . . 7 ⊢ 𝐹 ∈ ℕ0
26	25	nn0rei 12535	. . . . . 6 ⊢ 𝐹 ∈ ℝ
27	13	nnrei 12273	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
28	13	nngt0i 12303	. . . . . . 7 ⊢ 0 < 𝐸
29	27, 28	pm3.2i 470	. . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30		ledivmul 12142	. . . . . 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 1460	. . . . 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 5171	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
34	9	nnrei 12273	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ
35		log2ublem1.2	. . . . 5 ⊢ 𝐴 ∈ ℝ
36	34, 35	remulcli 11275	. . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
37	11	nn0rei 12535	. . . . . 6 ⊢ 𝐷 ∈ ℝ
38		nndivre 12305	. . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
39	37, 13, 38	mp2an 692	. . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ
40	34, 39	remulcli 11275	. . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41		log2ublem1.5	. . . . 5 ⊢ 𝐵 ∈ ℕ0
42	41	nn0rei 12535	. . . 4 ⊢ 𝐵 ∈ ℝ
43	36, 40, 42, 26	le2addi 11824	. . 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 692	. 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 7442	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
47	35	recni 11273	. . . 4 ⊢ 𝐴 ∈ ℂ
48	39	recni 11273	. . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ
49	10, 47, 48	adddii 11271	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
50	46, 49	eqtr2i 2764	. 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 5177	1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  (class class class)co 7431  ℝcr 11152  0cc0 11153   + caddc 11156   · cmul 11158   < clt 11293   ≤ cle 11294   / cdiv 11918  ℕcn 12264  3c3 12320  5c5 12322  7c7 12324  ℕ₀cn0 12524  ↑cexp 14099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-exp 14100

This theorem is referenced by:  log2ublem2  27005  log2ub  27007