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Mirrors > Home > MPE Home > Th. List > absproddvds | Structured version Visualization version GIF version |
Description: The absolute value of the product of the elements of a finite subset of the integers is divisible by each element of this subset. (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) |
Ref | Expression |
---|---|
absproddvds | ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absproddvds.f | . . . 4 ⊢ (𝜑 → 𝑍 ∈ Fin) | |
2 | absproddvds.s | . . . 4 ⊢ (𝜑 → 𝑍 ⊆ ℤ) | |
3 | 1, 2 | fproddvdsd 16297 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 𝑚 ∥ ∏𝑧 ∈ 𝑍 𝑧) |
4 | 2 | sselda 3978 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ) |
5 | 2 | sselda 3978 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ ℤ) |
6 | 1, 5 | fprodzcl 15916 | . . . . . . 7 ⊢ (𝜑 → ∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ) |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ) |
8 | dvdsabsb 16238 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ ∏𝑧 ∈ 𝑍 𝑧 ∈ ℤ) → (𝑚 ∥ ∏𝑧 ∈ 𝑍 𝑧 ↔ 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧))) | |
9 | 4, 7, 8 | syl2anc 583 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑚 ∥ ∏𝑧 ∈ 𝑍 𝑧 ↔ 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧))) |
10 | 9 | biimpd 228 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑚 ∥ ∏𝑧 ∈ 𝑍 𝑧 → 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧))) |
11 | 10 | ralimdva 3162 | . . 3 ⊢ (𝜑 → (∀𝑚 ∈ 𝑍 𝑚 ∥ ∏𝑧 ∈ 𝑍 𝑧 → ∀𝑚 ∈ 𝑍 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧))) |
12 | 3, 11 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧)) |
13 | absproddvds.p | . . . 4 ⊢ 𝑃 = (abs‘∏𝑧 ∈ 𝑍 𝑧) | |
14 | 13 | breq2i 5150 | . . 3 ⊢ (𝑚 ∥ 𝑃 ↔ 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧)) |
15 | 14 | ralbii 3088 | . 2 ⊢ (∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑃 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ (abs‘∏𝑧 ∈ 𝑍 𝑧)) |
16 | 12, 15 | sylibr 233 | 1 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 class class class wbr 5142 ‘cfv 6542 Fincfn 8953 ℤcz 12574 abscabs 15199 ∏cprod 15867 ∥ cdvds 16216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-prod 15868 df-dvds 16217 |
This theorem is referenced by: fissn0dvds 16575 |
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