![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2lgslem1a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgslem1a 25161. (Contributed by AV, 18-Jun-2021.) |
Ref | Expression |
---|---|
2lgslem1a2 | ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11419 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | 1 | rehalfcld 11317 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℝ) |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 / 2) ∈ ℝ) |
4 | id 22 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℤ) | |
5 | 2z 11447 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 2 ∈ ℤ) |
7 | 4, 6 | zmulcld 11526 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℤ) |
8 | 7 | zred 11520 | . . . 4 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℝ) |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 · 2) ∈ ℝ) |
10 | 2re 11128 | . . . . 5 ⊢ 2 ∈ ℝ | |
11 | 2pos 11150 | . . . . 5 ⊢ 0 < 2 | |
12 | 10, 11 | pm3.2i 470 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2)) |
14 | ltdiv1 10925 | . . 3 ⊢ (((𝑁 / 2) ∈ ℝ ∧ (𝐼 · 2) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝑁 / 2) < (𝐼 · 2) ↔ ((𝑁 / 2) / 2) < ((𝐼 · 2) / 2))) | |
15 | 3, 9, 13, 14 | syl3anc 1366 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) < (𝐼 · 2) ↔ ((𝑁 / 2) / 2) < ((𝐼 · 2) / 2))) |
16 | zcn 11420 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℂ) |
18 | 2cnne0 11280 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
19 | 18 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
20 | divdiv1 10774 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
21 | 17, 19, 19, 20 | syl3anc 1366 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
22 | 2t2e4 11215 | . . . . 5 ⊢ (2 · 2) = 4 | |
23 | 22 | oveq2i 6701 | . . . 4 ⊢ (𝑁 / (2 · 2)) = (𝑁 / 4) |
24 | 21, 23 | syl6eq 2701 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
25 | zcn 11420 | . . . . 5 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℂ) | |
26 | 25 | adantl 481 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℂ) |
27 | 2cnd 11131 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ∈ ℂ) | |
28 | 2ne0 11151 | . . . . 5 ⊢ 2 ≠ 0 | |
29 | 28 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ≠ 0) |
30 | 26, 27, 29 | divcan4d 10845 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝐼 · 2) / 2) = 𝐼) |
31 | 24, 30 | breq12d 4698 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (((𝑁 / 2) / 2) < ((𝐼 · 2) / 2) ↔ (𝑁 / 4) < 𝐼)) |
32 | 4re 11135 | . . . . 5 ⊢ 4 ∈ ℝ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
34 | 4ne0 11155 | . . . . 5 ⊢ 4 ≠ 0 | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
36 | 1, 33, 35 | redivcld 10891 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
37 | fllt 12647 | . . 3 ⊢ (((𝑁 / 4) ∈ ℝ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) | |
38 | 36, 37 | sylan 487 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) |
39 | 15, 31, 38 | 3bitrrd 295 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 · cmul 9979 < clt 10112 / cdiv 10722 2c2 11108 4c4 11110 ℤcz 11415 ⌊cfl 12631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-fl 12633 |
This theorem is referenced by: 2lgslem1a 25161 |
Copyright terms: Public domain | W3C validator |