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Mirrors > Home > MPE Home > Th. List > absprodnn | Structured version Visualization version GIF version |
Description: The absolute value of the product of the elements of a finite subset of the integers not containing 0 is a poitive integer. (Contributed by AV, 21-Aug-2020.) |
Ref | Expression |
---|---|
absproddvds.s | ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
absproddvds.f | ⊢ (𝜑 → 𝑍 ∈ Fin) |
absproddvds.p | ⊢ 𝑃 = (abs‘∏𝑧 ∈ 𝑍 𝑧) |
absprodnn.z | ⊢ (𝜑 → 0 ∉ 𝑍) |
Ref | Expression |
---|---|
absprodnn | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absproddvds.p | . 2 ⊢ 𝑃 = (abs‘∏𝑧 ∈ 𝑍 𝑧) | |
2 | absproddvds.f | . . . 4 ⊢ (𝜑 → 𝑍 ∈ Fin) | |
3 | absproddvds.s | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℤ) | |
4 | 3 | sselda 3821 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ ℤ) |
5 | 2, 4 | fprodzcl 15087 | . . 3 ⊢ (𝜑 → ∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ) |
6 | 4 | zcnd 11835 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ ℂ) |
7 | absprodnn.z | . . . . . 6 ⊢ (𝜑 → 0 ∉ 𝑍) | |
8 | elnelne2 3086 | . . . . . . 7 ⊢ ((𝑧 ∈ 𝑍 ∧ 0 ∉ 𝑍) → 𝑧 ≠ 0) | |
9 | 8 | expcom 404 | . . . . . 6 ⊢ (0 ∉ 𝑍 → (𝑧 ∈ 𝑍 → 𝑧 ≠ 0)) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑍 → 𝑧 ≠ 0)) |
11 | 10 | imp 397 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ≠ 0) |
12 | 2, 6, 11 | fprodn0 15112 | . . 3 ⊢ (𝜑 → ∏𝑧 ∈ 𝑍 𝑧 ≠ 0) |
13 | nnabscl 14472 | . . 3 ⊢ ((∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ ∧ ∏𝑧 ∈ 𝑍 𝑧 ≠ 0) → (abs‘∏𝑧 ∈ 𝑍 𝑧) ∈ ℕ) | |
14 | 5, 12, 13 | syl2anc 579 | . 2 ⊢ (𝜑 → (abs‘∏𝑧 ∈ 𝑍 𝑧) ∈ ℕ) |
15 | 1, 14 | syl5eqel 2863 | 1 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∉ wnel 3075 ⊆ wss 3792 ‘cfv 6135 Fincfn 8241 0cc0 10272 ℕcn 11374 ℤcz 11728 abscabs 14381 ∏cprod 15038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-prod 15039 |
This theorem is referenced by: fissn0dvds 15738 |
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