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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 15949 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 4128 | . . 3 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ) | |
3 | eldifi 4122 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) |
5 | eldifi 4122 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ) | |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
7 | 4, 6 | mulcld 11258 | . . . . 5 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ ℂ) |
8 | eldifsni 4789 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0) | |
9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0) |
10 | eldifsni 4789 | . . . . . . . . 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 11890 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ≠ 0) |
13 | 12 | neneqd 2940 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → ¬ (𝑥 · 𝑦) = 0) |
14 | ovex 7447 | . . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ V | |
15 | 14 | elsn 4639 | . . . . . 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 3955 | . . . 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 2940 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 = 0) |
23 | elsng 4638 | . . . . . 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 3955 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (ℂ ∖ {0})) |
27 | ax-1cn 11190 | . . . . 5 ⊢ 1 ∈ ℂ | |
28 | ax-1ne0 11201 | . . . . . 6 ⊢ 1 ≠ 0 | |
29 | 1ex 11234 | . . . . . . 7 ⊢ 1 ∈ V | |
30 | 29 | elsn 4639 | . . . . . 6 ⊢ (1 ∈ {0} ↔ 1 = 0) |
31 | 28, 30 | nemtbir 3033 | . . . . 5 ⊢ ¬ 1 ∈ {0} |
32 | eldif 3954 | . . . . 5 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ {0})) | |
33 | 27, 31, 32 | mpbir2an 710 | . . . 4 ⊢ 1 ∈ (ℂ ∖ {0}) |
34 | 33 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (ℂ ∖ {0})) |
35 | 1, 2, 18, 19, 26, 34 | fprodcllemf 15928 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (ℂ ∖ {0})) |
36 | eldifsni 4789 | . 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 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 {csn 4624 (class class class)co 7414 Fincfn 8957 ℂcc 11130 0cc0 11132 1c1 11133 · cmul 11137 ∏cprod 15875 |
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 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
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 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-prod 15876 |
This theorem is referenced by: fprodle 15966 |
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