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Mirrors > Home > ILE Home > Th. List > fprodeq0g | GIF version |
Description: Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodeq0g.kph | ⊢ Ⅎ𝑘𝜑 |
fprodeq0g.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodeq0g.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodeq0g.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
fprodeq0g.b0 | ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 0) |
Ref | Expression |
---|---|
fprodeq0g | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodeq0g.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcvd 2307 | . . 3 ⊢ (𝜑 → Ⅎ𝑘0) | |
3 | fprodeq0g.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
4 | fprodeq0g.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
5 | fprodeq0g.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
6 | fprodeq0g.b0 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 0) | |
7 | 1, 2, 3, 4, 5, 6 | fprodsplit1f 11569 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
8 | diffisn 6853 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ∈ Fin) | |
9 | 3, 5, 8 | syl2anc 409 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ∈ Fin) |
10 | eldifi 3242 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝐶}) → 𝑘 ∈ 𝐴) | |
11 | 10, 4 | sylan2 284 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝐵 ∈ ℂ) |
12 | 1, 9, 11 | fprodclf 11570 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 ∈ ℂ) |
13 | 12 | mul02d 8284 | . 2 ⊢ (𝜑 → (0 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = 0) |
14 | 7, 13 | eqtrd 2197 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 ∖ cdif 3111 {csn 3573 (class class class)co 5839 Fincfn 6700 ℂcc 7745 0cc0 7747 · cmul 7752 ∏cprod 11485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 ax-arch 7866 ax-caucvg 7867 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-if 3519 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-isom 5194 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-frec 6353 df-1o 6378 df-oadd 6382 df-er 6495 df-en 6701 df-dom 6702 df-fin 6703 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 df-2 8910 df-3 8911 df-4 8912 df-n0 9109 df-z 9186 df-uz 9461 df-q 9552 df-rp 9584 df-fz 9939 df-fzo 10072 df-seqfrec 10375 df-exp 10449 df-ihash 10683 df-cj 10778 df-re 10779 df-im 10780 df-rsqrt 10934 df-abs 10935 df-clim 11214 df-proddc 11486 |
This theorem is referenced by: (None) |
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