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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 3972 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ ℤ) |
5 | 2, 4 | fprodzcl 15930 | . . 3 ⊢ (𝜑 → ∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ) |
6 | 4 | zcnd 12697 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ ℂ) |
7 | absprodnn.z | . . . . . 6 ⊢ (𝜑 → 0 ∉ 𝑍) | |
8 | elnelne2 3048 | . . . . . . 7 ⊢ ((𝑧 ∈ 𝑍 ∧ 0 ∉ 𝑍) → 𝑧 ≠ 0) | |
9 | 8 | expcom 412 | . . . . . 6 ⊢ (0 ∉ 𝑍 → (𝑧 ∈ 𝑍 → 𝑧 ≠ 0)) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑍 → 𝑧 ≠ 0)) |
11 | 10 | imp 405 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ≠ 0) |
12 | 2, 6, 11 | fprodn0 15955 | . . 3 ⊢ (𝜑 → ∏𝑧 ∈ 𝑍 𝑧 ≠ 0) |
13 | nnabscl 15304 | . . 3 ⊢ ((∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ ∧ ∏𝑧 ∈ 𝑍 𝑧 ≠ 0) → (abs‘∏𝑧 ∈ 𝑍 𝑧) ∈ ℕ) | |
14 | 5, 12, 13 | syl2anc 582 | . 2 ⊢ (𝜑 → (abs‘∏𝑧 ∈ 𝑍 𝑧) ∈ ℕ) |
15 | 1, 14 | eqeltrid 2829 | 1 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∉ wnel 3036 ⊆ wss 3939 ‘cfv 6543 Fincfn 8962 0cc0 11138 ℕcn 12242 ℤcz 12588 abscabs 15213 ∏cprod 15881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-prod 15882 |
This theorem is referenced by: fissn0dvds 16589 |
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