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Mirrors > Home > MPE Home > Th. List > 2lgslem1a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgslem1a 25530. (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 11709 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | 1 | rehalfcld 11606 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℝ) |
3 | 2 | adantr 474 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 / 2) ∈ ℝ) |
4 | id 22 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℤ) | |
5 | 2z 11738 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 2 ∈ ℤ) |
7 | 4, 6 | zmulcld 11817 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℤ) |
8 | 7 | zred 11811 | . . . 4 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℝ) |
9 | 8 | adantl 475 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 · 2) ∈ ℝ) |
10 | 2re 11426 | . . . . 5 ⊢ 2 ∈ ℝ | |
11 | 2pos 11462 | . . . . 5 ⊢ 0 < 2 | |
12 | 10, 11 | pm3.2i 464 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2)) |
14 | ltdiv1 11218 | . . 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 1496 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) < (𝐼 · 2) ↔ ((𝑁 / 2) / 2) < ((𝐼 · 2) / 2))) |
16 | zcn 11710 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | adantr 474 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℂ) |
18 | 2cnne0 11569 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
19 | 18 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
20 | divdiv1 11063 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
21 | 17, 19, 19, 20 | syl3anc 1496 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
22 | 2t2e4 11523 | . . . . 5 ⊢ (2 · 2) = 4 | |
23 | 22 | oveq2i 6917 | . . . 4 ⊢ (𝑁 / (2 · 2)) = (𝑁 / 4) |
24 | 21, 23 | syl6eq 2878 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
25 | zcn 11710 | . . . . 5 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℂ) | |
26 | 25 | adantl 475 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℂ) |
27 | 2cnd 11430 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ∈ ℂ) | |
28 | 2ne0 11463 | . . . . 5 ⊢ 2 ≠ 0 | |
29 | 28 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ≠ 0) |
30 | 26, 27, 29 | divcan4d 11134 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝐼 · 2) / 2) = 𝐼) |
31 | 24, 30 | breq12d 4887 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (((𝑁 / 2) / 2) < ((𝐼 · 2) / 2) ↔ (𝑁 / 4) < 𝐼)) |
32 | 4re 11437 | . . . . 5 ⊢ 4 ∈ ℝ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
34 | 4ne0 11467 | . . . . 5 ⊢ 4 ≠ 0 | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
36 | 1, 33, 35 | redivcld 11180 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
37 | fllt 12903 | . . 3 ⊢ (((𝑁 / 4) ∈ ℝ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) | |
38 | 36, 37 | sylan 577 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) |
39 | 15, 31, 38 | 3bitrrd 298 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 ℝcr 10252 0cc0 10253 · cmul 10258 < clt 10392 / cdiv 11010 2c2 11407 4c4 11409 ℤcz 11705 ⌊cfl 12887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-n0 11620 df-z 11706 df-uz 11970 df-fl 12889 |
This theorem is referenced by: 2lgslem1a 25530 |
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