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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 14548 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 3700 | . . 3 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ) | |
3 | eldifi 3694 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) |
5 | eldifi 3694 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ) | |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
7 | 4, 6 | mulcld 9939 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ ℂ) |
8 | eldifsni 4261 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0) | |
9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0) |
10 | eldifsni 4261 | . . . . . . . . 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 10558 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ≠ 0) |
13 | 12 | neneqd 2787 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) = 0) |
14 | ovex 6577 | . . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ V | |
15 | 14 | elsn 4140 | . . . . . 6 ⊢ ((𝑥 · 𝑦) ∈ {0} ↔ (𝑥 · 𝑦) = 0) |
16 | 13, 15 | sylnibr 318 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) ∈ {0}) |
17 | 7, 16 | eldifd 3551 | . . . 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 2787 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 = 0) |
23 | elsng 4139 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ {0} ↔ 𝐵 = 0)) | |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ {0} ↔ 𝐵 = 0)) |
25 | 22, 24 | mtbird 314 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 ∈ {0}) |
26 | 20, 25 | eldifd 3551 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (ℂ ∖ {0})) |
27 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
28 | 0ne1 10965 | . . . . . . . . 9 ⊢ 0 ≠ 1 | |
29 | 28 | necomi 2836 | . . . . . . . 8 ⊢ 1 ≠ 0 |
30 | neneq 2788 | . . . . . . . 8 ⊢ (1 ≠ 0 → ¬ 1 = 0) | |
31 | 29, 30 | ax-mp 5 | . . . . . . 7 ⊢ ¬ 1 = 0 |
32 | 1ex 9914 | . . . . . . . 8 ⊢ 1 ∈ V | |
33 | 32 | elsn 4140 | . . . . . . 7 ⊢ (1 ∈ {0} ↔ 1 = 0) |
34 | 31, 33 | mtbir 312 | . . . . . 6 ⊢ ¬ 1 ∈ {0} |
35 | 27, 34 | pm3.2i 470 | . . . . 5 ⊢ (1 ∈ ℂ ∧ ¬ 1 ∈ {0}) |
36 | eldif 3550 | . . . . 5 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
37 | 35, 36 | mpbir 220 | . . . 4 ⊢ 1 ∈ (ℂ ∖ {0}) |
38 | 37 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (ℂ ∖ {0})) |
39 | 1, 2, 18, 19, 26, 38 | fprodcllemf 14527 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0})) |
40 | eldifsni 4261 | . 2 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0}) → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0) | |
41 | 39, 40 | syl 17 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 |
This theorem is referenced by: fprodle 14566 |
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