Theorem log2ublem1 24673

Description: Lemma for log2ub 24676. The proof of log2ub 24676, which is simply the evaluation of log2tlbnd 24672 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 11186	. . . . . . . 8 ⊢ 3 ∈ ℕ
3		7nn0 11314	. . . . . . . 8 ⊢ 7 ∈ ℕ0
4		nnexpcl 12873	. . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
5	2, 3, 4	mp2an 708	. . . . . . 7 ⊢ (3↑7) ∈ ℕ
6		5nn 11188	. . . . . . . 8 ⊢ 5 ∈ ℕ
7		7nn 11190	. . . . . . . 8 ⊢ 7 ∈ ℕ
8	6, 7	nnmulcli 11044	. . . . . . 7 ⊢ (5 · 7) ∈ ℕ
9	5, 8	nnmulcli 11044	. . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ
10	9	nncni 11030	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ
11		log2ublem1.3	. . . . . 6 ⊢ 𝐷 ∈ ℕ0
12	11	nn0cni 11304	. . . . 5 ⊢ 𝐷 ∈ ℂ
13		log2ublem1.4	. . . . . 6 ⊢ 𝐸 ∈ ℕ
14	13	nncni 11030	. . . . 5 ⊢ 𝐸 ∈ ℂ
15	13	nnne0i 11055	. . . . 5 ⊢ 𝐸 ≠ 0
16	10, 12, 14, 15	divassi 10781	. . . 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 11310	. . . . . . . . . 10 ⊢ 3 ∈ ℕ0
19	18, 3	nn0expcli 12886	. . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0
20		5nn0 11312	. . . . . . . . . 10 ⊢ 5 ∈ ℕ0
21	20, 3	nn0mulcli 11331	. . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0
22	19, 21	nn0mulcli 11331	. . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0
23	22, 11	nn0mulcli 11331	. . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
24	23	nn0rei 11303	. . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25		log2ublem1.6	. . . . . . 7 ⊢ 𝐹 ∈ ℕ0
26	25	nn0rei 11303	. . . . . 6 ⊢ 𝐹 ∈ ℝ
27	13	nnrei 11029	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
28	13	nngt0i 11054	. . . . . . 7 ⊢ 0 < 𝐸
29	27, 28	pm3.2i 471	. . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30		ledivmul 10899	. . . . . 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 1424	. . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
32	17, 31	mpbir 221	. . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
33	16, 32	eqbrtrri 4676	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
34	9	nnrei 11029	. . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ
35		log2ublem1.2	. . . . 5 ⊢ 𝐴 ∈ ℝ
36	34, 35	remulcli 10054	. . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
37	11	nn0rei 11303	. . . . . 6 ⊢ 𝐷 ∈ ℝ
38		nndivre 11056	. . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
39	37, 13, 38	mp2an 708	. . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ
40	34, 39	remulcli 10054	. . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41		log2ublem1.5	. . . . 5 ⊢ 𝐵 ∈ ℕ0
42	41	nn0rei 11303	. . . 4 ⊢ 𝐵 ∈ ℝ
43	36, 40, 42, 26	le2addi 10591	. . 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 708	. 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 6661	. . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
47	35	recni 10052	. . . 4 ⊢ 𝐴 ∈ ℂ
48	39	recni 10052	. . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ
49	10, 47, 48	adddii 10050	. . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
50	46, 49	eqtr2i 2645	. 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 4682	1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   class class class wbr 4653  (class class class)co 6650  ℝcr 9935  0cc0 9936   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   / cdiv 10684  ℕcn 11020  3c3 11071  5c5 11073  7c7 11075  ℕ₀cn0 11292  ↑cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  log2ublem2  24674  log2ub  24676