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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 12453 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
3 | lcmineqlem14.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
4 | 3 | nnzd 12453 | . . 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 40035 | . . . . . 6 ⊢ (𝜑 → ((𝐵 · 𝐶) ∥ 𝐸 ↔ 𝐵 ∥ (𝐸 / 𝐶))) |
9 | 5, 8 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝐵 ∥ (𝐸 / 𝐶)) |
10 | dvdszrcl 15996 | . . . . 5 ⊢ (𝐵 ∥ (𝐸 / 𝐶) → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ (𝐸 / 𝐶) ∈ ℤ)) |
12 | 11 | simprd 495 | . . 3 ⊢ (𝜑 → (𝐸 / 𝐶) ∈ ℤ) |
13 | lcmineqlem14.9 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) | |
14 | 6 | nnzd 12453 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
15 | 2, 14 | zmulcld 12460 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℤ) |
16 | lcmineqlem14.4 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℕ) | |
17 | 16 | nnzd 12453 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) |
18 | 7 | nnzd 12453 | . . . . 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 16032 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐸) |
22 | 1, 6, 7 | nnproddivdvdsd 40035 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶) ∥ 𝐸 ↔ 𝐴 ∥ (𝐸 / 𝐶))) |
23 | 21, 22 | mpbid 231 | . . 3 ⊢ (𝜑 → 𝐴 ∥ (𝐸 / 𝐶)) |
24 | 2, 4, 12, 13, 23, 9 | coprmdvds2d 40036 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) ∥ (𝐸 / 𝐶)) |
25 | 1, 3 | nnmulcld 12054 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
26 | 25, 6, 7 | nnproddivdvdsd 40035 | . 2 ⊢ (𝜑 → (((𝐴 · 𝐵) · 𝐶) ∥ 𝐸 ↔ (𝐴 · 𝐵) ∥ (𝐸 / 𝐶))) |
27 | 24, 26 | mpbird 256 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 class class class wbr 5077 (class class class)co 7295 1c1 10900 · cmul 10904 / cdiv 11660 ℕcn 12001 ℤcz 12347 ∥ cdvds 15991 gcd cgcd 16229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-sup 9229 df-inf 9230 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-z 12348 df-uz 12611 df-rp 12759 df-fl 13540 df-mod 13618 df-seq 13750 df-exp 13811 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-dvds 15992 df-gcd 16230 |
This theorem is referenced by: lcmineqlem19 40081 |
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