![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodeq02 | Structured version Visualization version GIF version |
Description: If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
Ref | Expression |
---|---|
fprodeq02.1 | ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) |
fprodeq02.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodeq02.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodeq02.k | ⊢ (𝜑 → 𝐾 ∈ 𝐴) |
fprodeq02.c | ⊢ (𝜑 → 𝐶 = 0) |
Ref | Expression |
---|---|
fprodeq02 | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjdif 4379 | . . . 4 ⊢ ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅) |
3 | fprodeq02.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝐴) | |
4 | 3 | snssd 4702 | . . . . 5 ⊢ (𝜑 → {𝐾} ⊆ 𝐴) |
5 | undif 4388 | . . . . 5 ⊢ ({𝐾} ⊆ 𝐴 ↔ ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) | |
6 | 4, 5 | sylib 221 | . . . 4 ⊢ (𝜑 → ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) |
7 | 6 | eqcomd 2804 | . . 3 ⊢ (𝜑 → 𝐴 = ({𝐾} ∪ (𝐴 ∖ {𝐾}))) |
8 | fprodeq02.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
9 | fprodeq02.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
10 | 2, 7, 8, 9 | fprodsplit 15312 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
11 | fprodeq02.c | . . . . . 6 ⊢ (𝜑 → 𝐶 = 0) | |
12 | 0cnd 10623 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2890 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | fprodeq02.1 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) | |
15 | 14 | prodsn 15308 | . . . . 5 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
16 | 3, 13, 15 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
17 | 16, 11 | eqtrd 2833 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 0) |
18 | 17 | oveq1d 7150 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
19 | diffi 8734 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐾}) ∈ Fin) | |
20 | 8, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ∈ Fin) |
21 | difssd 4060 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ⊆ 𝐴) | |
22 | 21 | sselda 3915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝑘 ∈ 𝐴) |
23 | 22, 9 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
24 | 20, 23 | fprodcl 15298 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵 ∈ ℂ) |
25 | 24 | mul02d 10827 | . 2 ⊢ (𝜑 → (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = 0) |
26 | 10, 18, 25 | 3eqtrd 2837 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 · cmul 10531 ∏cprod 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252 |
This theorem is referenced by: fprodex01 30567 |
Copyright terms: Public domain | W3C validator |