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Mirrors > Home > MPE Home > Th. List > dchrisum0lem1a | Structured version Visualization version GIF version |
Description: Lemma for dchrisum0lem1 25250. (Contributed by Mario Carneiro, 7-Jun-2016.) |
Ref | Expression |
---|---|
dchrisum0lem1a | ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn 12408 | . . . . . . 7 ⊢ (𝐷 ∈ (1...(⌊‘𝑋)) → 𝐷 ∈ ℕ) | |
2 | 1 | adantl 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝐷 ∈ ℕ) |
3 | 2 | nnred 11073 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝐷 ∈ ℝ) |
4 | simpr 476 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → 𝑋 ∈ ℝ+) | |
5 | 4 | rpregt0d 11916 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → (𝑋 ∈ ℝ ∧ 0 < 𝑋)) |
6 | 5 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ∈ ℝ ∧ 0 < 𝑋)) |
7 | 6 | simpld 474 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝑋 ∈ ℝ) |
8 | 4 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝑋 ∈ ℝ+) |
9 | 8 | rpge0d 11914 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 0 ≤ 𝑋) |
10 | 4 | rpred 11910 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → 𝑋 ∈ ℝ) |
11 | fznnfl 12701 | . . . . . . 7 ⊢ (𝑋 ∈ ℝ → (𝐷 ∈ (1...(⌊‘𝑋)) ↔ (𝐷 ∈ ℕ ∧ 𝐷 ≤ 𝑋))) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → (𝐷 ∈ (1...(⌊‘𝑋)) ↔ (𝐷 ∈ ℕ ∧ 𝐷 ≤ 𝑋))) |
13 | 12 | simplbda 653 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝐷 ≤ 𝑋) |
14 | 3, 7, 7, 9, 13 | lemul2ad 11002 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 · 𝐷) ≤ (𝑋 · 𝑋)) |
15 | rpcn 11879 | . . . . . . 7 ⊢ (𝑋 ∈ ℝ+ → 𝑋 ∈ ℂ) | |
16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → 𝑋 ∈ ℂ) |
17 | 16 | sqvald 13045 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → (𝑋↑2) = (𝑋 · 𝑋)) |
18 | 17 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋↑2) = (𝑋 · 𝑋)) |
19 | 14, 18 | breqtrrd 4713 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 · 𝐷) ≤ (𝑋↑2)) |
20 | 2z 11447 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
21 | rpexpcl 12919 | . . . . . . 7 ⊢ ((𝑋 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝑋↑2) ∈ ℝ+) | |
22 | 4, 20, 21 | sylancl 695 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → (𝑋↑2) ∈ ℝ+) |
23 | 22 | rpred 11910 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ+) → (𝑋↑2) ∈ ℝ) |
24 | 23 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋↑2) ∈ ℝ) |
25 | 2 | nnrpd 11908 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝐷 ∈ ℝ+) |
26 | 7, 24, 25 | lemuldivd 11959 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → ((𝑋 · 𝐷) ≤ (𝑋↑2) ↔ 𝑋 ≤ ((𝑋↑2) / 𝐷))) |
27 | 19, 26 | mpbid 222 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → 𝑋 ≤ ((𝑋↑2) / 𝐷)) |
28 | nndivre 11094 | . . . 4 ⊢ (((𝑋↑2) ∈ ℝ ∧ 𝐷 ∈ ℕ) → ((𝑋↑2) / 𝐷) ∈ ℝ) | |
29 | 23, 1, 28 | syl2an 493 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → ((𝑋↑2) / 𝐷) ∈ ℝ) |
30 | flword2 12654 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ ((𝑋↑2) / 𝐷) ∈ ℝ ∧ 𝑋 ≤ ((𝑋↑2) / 𝐷)) → (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋))) | |
31 | 7, 29, 27, 30 | syl3anc 1366 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋))) |
32 | 27, 31 | jca 553 | 1 ⊢ (((𝜑 ∧ 𝑋 ∈ ℝ+) ∧ 𝐷 ∈ (1...(⌊‘𝑋))) → (𝑋 ≤ ((𝑋↑2) / 𝐷) ∧ (⌊‘((𝑋↑2) / 𝐷)) ∈ (ℤ≥‘(⌊‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 < clt 10112 ≤ cle 10113 / cdiv 10722 ℕcn 11058 2c2 11108 ℤcz 11415 ℤ≥cuz 11725 ℝ+crp 11870 ...cfz 12364 ⌊cfl 12631 ↑cexp 12900 |
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-1st 7210 df-2nd 7211 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-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fl 12633 df-seq 12842 df-exp 12901 |
This theorem is referenced by: dchrisum0lem1b 25249 dchrisum0lem1 25250 |
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