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Mirrors > Home > MPE Home > Th. List > fprodn0f | Structured version Visualization version GIF version |
Description: A finite product of nonzero terms is nonzero. A version of fprodn0 16011 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodn0f.kph | ⊢ Ⅎ𝑘𝜑 |
fprodn0f.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodn0f.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodn0f.bne0 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
fprodn0f | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodn0f.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | difssd 4146 | . . 3 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ) | |
3 | eldifi 4140 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) |
5 | eldifi 4140 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ) | |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
7 | 4, 6 | mulcld 11278 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ ℂ) |
8 | eldifsni 4794 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0) | |
9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0) |
10 | eldifsni 4794 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0) | |
11 | 10 | adantl 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0) |
12 | 4, 6, 9, 11 | mulne0d 11912 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ≠ 0) |
13 | 12 | neneqd 2942 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) = 0) |
14 | ovex 7463 | . . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ V | |
15 | 14 | elsn 4645 | . . . . . 6 ⊢ ((𝑥 · 𝑦) ∈ {0} ↔ (𝑥 · 𝑦) = 0) |
16 | 13, 15 | sylnibr 329 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) ∈ {0}) |
17 | 7, 16 | eldifd 3973 | . . . 4 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
18 | 17 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
19 | fprodn0f.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
20 | fprodn0f.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
21 | fprodn0f.bne0 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≠ 0) | |
22 | 21 | neneqd 2942 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 = 0) |
23 | elsng 4644 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ {0} ↔ 𝐵 = 0)) | |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ {0} ↔ 𝐵 = 0)) |
25 | 22, 24 | mtbird 325 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 ∈ {0}) |
26 | 20, 25 | eldifd 3973 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (ℂ ∖ {0})) |
27 | ax-1cn 11210 | . . . . 5 ⊢ 1 ∈ ℂ | |
28 | ax-1ne0 11221 | . . . . . 6 ⊢ 1 ≠ 0 | |
29 | 1ex 11254 | . . . . . . 7 ⊢ 1 ∈ V | |
30 | 29 | elsn 4645 | . . . . . 6 ⊢ (1 ∈ {0} ↔ 1 = 0) |
31 | 28, 30 | nemtbir 3035 | . . . . 5 ⊢ ¬ 1 ∈ {0} |
32 | eldif 3972 | . . . . 5 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
33 | 27, 31, 32 | mpbir2an 711 | . . . 4 ⊢ 1 ∈ (ℂ ∖ {0}) |
34 | 33 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (ℂ ∖ {0})) |
35 | 1, 2, 18, 19, 26, 34 | fprodcllemf 15990 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0})) |
36 | eldifsni 4794 | . 2 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0}) → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0) | |
37 | 35, 36 | syl 17 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 (class class class)co 7430 Fincfn 8983 ℂcc 11150 0cc0 11152 1c1 11153 · cmul 11157 ∏cprod 15935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-prod 15936 |
This theorem is referenced by: fprodle 16028 |
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