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 4371 | . . . 4 ⊢ ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅) |
3 | fprodeq02.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝐴) | |
4 | 3 | snssd 4707 | . . . . 5 ⊢ (𝜑 → {𝐾} ⊆ 𝐴) |
5 | undif 4381 | . . . . 5 ⊢ ({𝐾} ⊆ 𝐴 ↔ ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) | |
6 | 4, 5 | sylib 221 | . . . 4 ⊢ (𝜑 → ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) |
7 | 6 | eqcomd 2745 | . . 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 15424 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
11 | fprodeq02.c | . . . . . 6 ⊢ (𝜑 → 𝐶 = 0) | |
12 | 0cnd 10724 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2834 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | fprodeq02.1 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) | |
15 | 14 | prodsn 15420 | . . . . 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 2774 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 0) |
18 | 17 | oveq1d 7197 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
19 | diffi 8839 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐾}) ∈ Fin) | |
20 | 8, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ∈ Fin) |
21 | difssd 4033 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ⊆ 𝐴) | |
22 | 21 | sselda 3887 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝑘 ∈ 𝐴) |
23 | 22, 9 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
24 | 20, 23 | fprodcl 15410 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵 ∈ ℂ) |
25 | 24 | mul02d 10928 | . 2 ⊢ (𝜑 → (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = 0) |
26 | 10, 18, 25 | 3eqtrd 2778 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3850 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 {csn 4526 (class class class)co 7182 Fincfn 8567 ℂcc 10625 0cc0 10627 · cmul 10632 ∏cprod 15363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-sup 8991 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-fz 12994 df-fzo 13137 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-prod 15364 |
This theorem is referenced by: fprodex01 30726 |
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