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Mathbox for Thierry Arnoux |
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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 4475 | . . . 4 ⊢ ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅) |
3 | fprodeq02.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝐴) | |
4 | 3 | snssd 4817 | . . . . 5 ⊢ (𝜑 → {𝐾} ⊆ 𝐴) |
5 | undif 4485 | . . . . 5 ⊢ ({𝐾} ⊆ 𝐴 ↔ ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝜑 → ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) |
7 | 6 | eqcomd 2734 | . . 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 15950 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
11 | fprodeq02.c | . . . . . 6 ⊢ (𝜑 → 𝐶 = 0) | |
12 | 0cnd 11245 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2829 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | fprodeq02.1 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) | |
15 | 14 | prodsn 15946 | . . . . 5 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
16 | 3, 13, 15 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
17 | 16, 11 | eqtrd 2768 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 0) |
18 | 17 | oveq1d 7441 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
19 | diffi 9210 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐾}) ∈ Fin) | |
20 | 8, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ∈ Fin) |
21 | difssd 4133 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ⊆ 𝐴) | |
22 | 21 | sselda 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝑘 ∈ 𝐴) |
23 | 22, 9 | syldan 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
24 | 20, 23 | fprodcl 15936 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵 ∈ ℂ) |
25 | 24 | mul02d 11450 | . 2 ⊢ (𝜑 → (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = 0) |
26 | 10, 18, 25 | 3eqtrd 2772 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 {csn 4632 (class class class)co 7426 Fincfn 8970 ℂcc 11144 0cc0 11146 · cmul 11151 ∏cprod 15889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-prod 15890 |
This theorem is referenced by: fprodex01 32609 |
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