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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 15335 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 4111 | . . 3 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ) | |
3 | eldifi 4105 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ) | |
4 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) |
5 | eldifi 4105 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ) | |
6 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
7 | 4, 6 | mulcld 10663 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ ℂ) |
8 | eldifsni 4724 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0) | |
9 | 8 | adantr 483 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0) |
10 | eldifsni 4724 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0) | |
11 | 10 | adantl 484 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0) |
12 | 4, 6, 9, 11 | mulne0d 11294 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ≠ 0) |
13 | 12 | neneqd 3023 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) = 0) |
14 | ovex 7191 | . . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ V | |
15 | 14 | elsn 4584 | . . . . . 6 ⊢ ((𝑥 · 𝑦) ∈ {0} ↔ (𝑥 · 𝑦) = 0) |
16 | 13, 15 | sylnibr 331 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) ∈ {0}) |
17 | 7, 16 | eldifd 3949 | . . . 4 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
18 | 17 | adantl 484 | . . 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 3023 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 = 0) |
23 | elsng 4583 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ {0} ↔ 𝐵 = 0)) | |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ {0} ↔ 𝐵 = 0)) |
25 | 22, 24 | mtbird 327 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 ∈ {0}) |
26 | 20, 25 | eldifd 3949 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (ℂ ∖ {0})) |
27 | ax-1cn 10597 | . . . . 5 ⊢ 1 ∈ ℂ | |
28 | ax-1ne0 10608 | . . . . . 6 ⊢ 1 ≠ 0 | |
29 | 1ex 10639 | . . . . . . 7 ⊢ 1 ∈ V | |
30 | 29 | elsn 4584 | . . . . . 6 ⊢ (1 ∈ {0} ↔ 1 = 0) |
31 | 28, 30 | nemtbir 3114 | . . . . 5 ⊢ ¬ 1 ∈ {0} |
32 | eldif 3948 | . . . . 5 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
33 | 27, 31, 32 | mpbir2an 709 | . . . 4 ⊢ 1 ∈ (ℂ ∖ {0}) |
34 | 33 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (ℂ ∖ {0})) |
35 | 1, 2, 18, 19, 26, 34 | fprodcllemf 15314 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0})) |
36 | eldifsni 4724 | . 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 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 ∏cprod 15261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 |
This theorem is referenced by: fprodle 15352 |
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