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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmineqlem14 | Structured version Visualization version GIF version |
Description: Technical lemma for inequality estimate. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
lcmineqlem14.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
lcmineqlem14.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
lcmineqlem14.3 | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
lcmineqlem14.4 | ⊢ (𝜑 → 𝐷 ∈ ℕ) |
lcmineqlem14.5 | ⊢ (𝜑 → 𝐸 ∈ ℕ) |
lcmineqlem14.6 | ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐷) |
lcmineqlem14.7 | ⊢ (𝜑 → (𝐵 · 𝐶) ∥ 𝐸) |
lcmineqlem14.8 | ⊢ (𝜑 → 𝐷 ∥ 𝐸) |
lcmineqlem14.9 | ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) |
Ref | Expression |
---|---|
lcmineqlem14 | ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmineqlem14.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nnzd 12160 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
3 | lcmineqlem14.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnzd 12160 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
5 | lcmineqlem14.7 | . . . . . 6 ⊢ (𝜑 → (𝐵 · 𝐶) ∥ 𝐸) | |
6 | lcmineqlem14.3 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
7 | lcmineqlem14.5 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ) | |
8 | 3, 6, 7 | nnproddivdvdsd 39618 | . . . . . 6 ⊢ (𝜑 → ((𝐵 · 𝐶) ∥ 𝐸 ↔ 𝐵 ∥ (𝐸 / 𝐶))) |
9 | 5, 8 | mpbid 235 | . . . . 5 ⊢ (𝜑 → 𝐵 ∥ (𝐸 / 𝐶)) |
10 | dvdszrcl 15697 | . . . . 5 ⊢ (𝐵 ∥ (𝐸 / 𝐶) → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) |
12 | 11 | simprd 499 | . . 3 ⊢ (𝜑 → (𝐸 / 𝐶) ∈ ℤ) |
13 | lcmineqlem14.9 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | |
14 | 6 | nnzd 12160 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
15 | 2, 14 | zmulcld 12167 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℤ) |
16 | lcmineqlem14.4 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℕ) | |
17 | 16 | nnzd 12160 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) |
18 | 7 | nnzd 12160 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℤ) |
19 | lcmineqlem14.6 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐷) | |
20 | lcmineqlem14.8 | . . . . 5 ⊢ (𝜑 → 𝐷 ∥ 𝐸) | |
21 | 15, 17, 18, 19, 20 | dvdstrd 15733 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐸) |
22 | 1, 6, 7 | nnproddivdvdsd 39618 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶) ∥ 𝐸 ↔ 𝐴 ∥ (𝐸 / 𝐶))) |
23 | 21, 22 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝐴 ∥ (𝐸 / 𝐶)) |
24 | 2, 4, 12, 13, 23, 9 | coprmdvds2d 39619 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) ∥ (𝐸 / 𝐶)) |
25 | 1, 3 | nnmulcld 11762 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
26 | 25, 6, 7 | nnproddivdvdsd 39618 | . 2 ⊢ (𝜑 → (((𝐴 · 𝐵) · 𝐶) ∥ 𝐸 ↔ (𝐴 · 𝐵) ∥ (𝐸 / 𝐶))) |
27 | 24, 26 | mpbird 260 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 (class class class)co 7164 1c1 10609 · cmul 10613 / cdiv 11368 ℕcn 11709 ℤcz 12055 ∥ cdvds 15692 gcd cgcd 15930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-fl 13246 df-mod 13322 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-dvds 15693 df-gcd 15931 |
This theorem is referenced by: lcmineqlem19 39664 |
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