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Mirrors > Home > MPE Home > Th. List > 2lgslem1a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgslem1a 26244. (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 12163 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | 1 | rehalfcld 12060 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℝ) |
3 | 2 | adantr 484 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 / 2) ∈ ℝ) |
4 | id 22 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℤ) | |
5 | 2z 12192 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 2 ∈ ℤ) |
7 | 4, 6 | zmulcld 12271 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℤ) |
8 | 7 | zred 12265 | . . . 4 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℝ) |
9 | 8 | adantl 485 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 · 2) ∈ ℝ) |
10 | 2re 11887 | . . . . 5 ⊢ 2 ∈ ℝ | |
11 | 2pos 11916 | . . . . 5 ⊢ 0 < 2 | |
12 | 10, 11 | pm3.2i 474 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2)) |
14 | ltdiv1 11679 | . . 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 1373 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) < (𝐼 · 2) ↔ ((𝑁 / 2) / 2) < ((𝐼 · 2) / 2))) |
16 | zcn 12164 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℂ) |
18 | 2cnne0 12023 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
19 | 18 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
20 | divdiv1 11526 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
21 | 17, 19, 19, 20 | syl3anc 1373 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
22 | 2t2e4 11977 | . . . . 5 ⊢ (2 · 2) = 4 | |
23 | 22 | oveq2i 7213 | . . . 4 ⊢ (𝑁 / (2 · 2)) = (𝑁 / 4) |
24 | 21, 23 | eqtrdi 2790 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
25 | zcn 12164 | . . . . 5 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℂ) | |
26 | 25 | adantl 485 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℂ) |
27 | 2cnd 11891 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ∈ ℂ) | |
28 | 2ne0 11917 | . . . . 5 ⊢ 2 ≠ 0 | |
29 | 28 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ≠ 0) |
30 | 26, 27, 29 | divcan4d 11597 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝐼 · 2) / 2) = 𝐼) |
31 | 24, 30 | breq12d 5056 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (((𝑁 / 2) / 2) < ((𝐼 · 2) / 2) ↔ (𝑁 / 4) < 𝐼)) |
32 | 4re 11897 | . . . . 5 ⊢ 4 ∈ ℝ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
34 | 4ne0 11921 | . . . . 5 ⊢ 4 ≠ 0 | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
36 | 1, 33, 35 | redivcld 11643 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
37 | fllt 13364 | . . 3 ⊢ (((𝑁 / 4) ∈ ℝ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) | |
38 | 36, 37 | sylan 583 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) |
39 | 15, 31, 38 | 3bitrrd 309 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 · cmul 10717 < clt 10850 / cdiv 11472 2c2 11868 4c4 11870 ℤcz 12159 ⌊cfl 13348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-n0 12074 df-z 12160 df-uz 12422 df-fl 13350 |
This theorem is referenced by: 2lgslem1a 26244 |
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