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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 12110 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
3 | lcmineqlem14.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnzd 12110 | . . 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 39553 | . . . . . 6 ⊢ (𝜑 → ((𝐵 · 𝐶) ∥ 𝐸 ↔ 𝐵 ∥ (𝐸 / 𝐶))) |
9 | 5, 8 | mpbid 235 | . . . . 5 ⊢ (𝜑 → 𝐵 ∥ (𝐸 / 𝐶)) |
10 | dvdszrcl 15645 | . . . . 5 ⊢ (𝐵 ∥ (𝐸 / 𝐶) → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) |
12 | 11 | simprd 500 | . . 3 ⊢ (𝜑 → (𝐸 / 𝐶) ∈ ℤ) |
13 | lcmineqlem14.9 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | |
14 | 6 | nnzd 12110 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
15 | 2, 14 | zmulcld 12117 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℤ) |
16 | lcmineqlem14.4 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℕ) | |
17 | 16 | nnzd 12110 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) |
18 | 7 | nnzd 12110 | . . . . 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 15681 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐸) |
22 | 1, 6, 7 | nnproddivdvdsd 39553 | . . . 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 39554 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) ∥ (𝐸 / 𝐶)) |
25 | 1, 3 | nnmulcld 11712 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
26 | 25, 6, 7 | nnproddivdvdsd 39553 | . 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 400 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 (class class class)co 7143 1c1 10561 · cmul 10565 / cdiv 11320 ℕcn 11659 ℤcz 12005 ∥ cdvds 15640 gcd cgcd 15878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-sup 8924 df-inf 8925 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-n0 11920 df-z 12006 df-uz 12268 df-rp 12416 df-fl 13196 df-mod 13272 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-dvds 15641 df-gcd 15879 |
This theorem is referenced by: lcmineqlem19 39599 |
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