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 4423 | . . . 4 ⊢ ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐾} ∩ (𝐴 ∖ {𝐾})) = ∅) |
3 | fprodeq02.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝐴) | |
4 | 3 | snssd 4744 | . . . . 5 ⊢ (𝜑 → {𝐾} ⊆ 𝐴) |
5 | undif 4432 | . . . . 5 ⊢ ({𝐾} ⊆ 𝐴 ↔ ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) | |
6 | 4, 5 | sylib 220 | . . . 4 ⊢ (𝜑 → ({𝐾} ∪ (𝐴 ∖ {𝐾})) = 𝐴) |
7 | 6 | eqcomd 2829 | . . 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 15322 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
11 | fprodeq02.c | . . . . . 6 ⊢ (𝜑 → 𝐶 = 0) | |
12 | 0cnd 10636 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2915 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | fprodeq02.1 | . . . . . 6 ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) | |
15 | 14 | prodsn 15318 | . . . . 5 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
16 | 3, 13, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 𝐶) |
17 | 16, 11 | eqtrd 2858 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐾}𝐵 = 0) |
18 | 17 | oveq1d 7173 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐾}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵)) |
19 | diffi 8752 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐾}) ∈ Fin) | |
20 | 8, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ∈ Fin) |
21 | difssd 4111 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ {𝐾}) ⊆ 𝐴) | |
22 | 21 | sselda 3969 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝑘 ∈ 𝐴) |
23 | 22, 9 | syldan 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐾})) → 𝐵 ∈ ℂ) |
24 | 20, 23 | fprodcl 15308 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵 ∈ ℂ) |
25 | 24 | mul02d 10840 | . 2 ⊢ (𝜑 → (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐾})𝐵) = 0) |
26 | 10, 18, 25 | 3eqtrd 2862 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 · cmul 10544 ∏cprod 15261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 |
This theorem is referenced by: fprodex01 30543 |
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