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Mirrors > Home > MPE Home > Th. List > 2lgslem1a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2lgslem1a 25970. (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 11988 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | 1 | rehalfcld 11887 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℝ) |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 / 2) ∈ ℝ) |
4 | id 22 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℤ) | |
5 | 2z 12017 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → 2 ∈ ℤ) |
7 | 4, 6 | zmulcld 12096 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℤ) |
8 | 7 | zred 12090 | . . . 4 ⊢ (𝐼 ∈ ℤ → (𝐼 · 2) ∈ ℝ) |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 · 2) ∈ ℝ) |
10 | 2re 11714 | . . . . 5 ⊢ 2 ∈ ℝ | |
11 | 2pos 11743 | . . . . 5 ⊢ 0 < 2 | |
12 | 10, 11 | pm3.2i 473 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℝ ∧ 0 < 2)) |
14 | ltdiv1 11507 | . . 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 1367 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) < (𝐼 · 2) ↔ ((𝑁 / 2) / 2) < ((𝐼 · 2) / 2))) |
16 | zcn 11989 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℂ) |
18 | 2cnne0 11850 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
19 | 18 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
20 | divdiv1 11354 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
21 | 17, 19, 19, 20 | syl3anc 1367 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
22 | 2t2e4 11804 | . . . . 5 ⊢ (2 · 2) = 4 | |
23 | 22 | oveq2i 7170 | . . . 4 ⊢ (𝑁 / (2 · 2)) = (𝑁 / 4) |
24 | 21, 23 | syl6eq 2875 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
25 | zcn 11989 | . . . . 5 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℂ) | |
26 | 25 | adantl 484 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℂ) |
27 | 2cnd 11718 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ∈ ℂ) | |
28 | 2ne0 11744 | . . . . 5 ⊢ 2 ≠ 0 | |
29 | 28 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 2 ≠ 0) |
30 | 26, 27, 29 | divcan4d 11425 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝐼 · 2) / 2) = 𝐼) |
31 | 24, 30 | breq12d 5082 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (((𝑁 / 2) / 2) < ((𝐼 · 2) / 2) ↔ (𝑁 / 4) < 𝐼)) |
32 | 4re 11724 | . . . . 5 ⊢ 4 ∈ ℝ | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
34 | 4ne0 11748 | . . . . 5 ⊢ 4 ≠ 0 | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
36 | 1, 33, 35 | redivcld 11471 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
37 | fllt 13179 | . . 3 ⊢ (((𝑁 / 4) ∈ ℝ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) | |
38 | 36, 37 | sylan 582 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝑁 / 4) < 𝐼 ↔ (⌊‘(𝑁 / 4)) < 𝐼)) |
39 | 15, 31, 38 | 3bitrrd 308 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((⌊‘(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 · 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 · cmul 10545 < clt 10678 / cdiv 11300 2c2 11695 4c4 11697 ℤcz 11984 ⌊cfl 13163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-fl 13165 |
This theorem is referenced by: 2lgslem1a 25970 |
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