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Mirrors > Home > ILE Home > Th. List > fprodsplitsn | GIF version |
Description: Separate out a term in a finite product. See also fprodunsn 11505 which is the same but with a distinct variable condition in place of Ⅎ𝑘𝜑. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodsplitsn.ph | ⊢ Ⅎ𝑘𝜑 |
fprodsplitsn.kd | ⊢ Ⅎ𝑘𝐷 |
fprodsplitsn.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodsplitsn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fprodsplitsn.ba | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) |
fprodsplitsn.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
fprodsplitsn.d | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) |
fprodsplitsn.dcn | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
fprodsplitsn | ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodsplitsn.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | fprodsplitsn.ba | . . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | |
3 | disjsn 3622 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 133 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2158 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fprodsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | fprodsplitsn.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
8 | snfig 6760 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ∈ Fin) | |
9 | 7, 8 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐵} ∈ Fin) |
10 | unfidisj 6867 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin ∧ (𝐴 ∩ {𝐵}) = ∅) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
11 | 6, 9, 4, 10 | syl3anc 1220 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
12 | fprodsplitsn.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
13 | 12 | ex 114 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
14 | fprodsplitsn.d | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
15 | 14 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
16 | fprodsplitsn.dcn | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
17 | 16 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
18 | 15, 17 | eqeltrd 2234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
19 | 18 | ex 114 | . . . . 5 ⊢ (𝜑 → (𝑘 = 𝐵 → 𝐶 ∈ ℂ)) |
20 | 13, 19 | jaod 707 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵) → 𝐶 ∈ ℂ)) |
21 | elun 3248 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵})) | |
22 | elsni 3578 | . . . . . 6 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
23 | 22 | orim2i 751 | . . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵)) |
24 | 21, 23 | sylbi 120 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 = 𝐵)) |
25 | 20, 24 | impel 278 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
26 | 1, 4, 5, 11, 25 | fprodsplitf 11533 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶)) |
27 | fprodsplitsn.kd | . . . . 5 ⊢ Ⅎ𝑘𝐷 | |
28 | 27, 14 | prodsnf 11493 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
29 | 7, 16, 28 | syl2anc 409 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
30 | 29 | oveq2d 5841 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
31 | 26, 30 | eqtrd 2190 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1335 Ⅎwnf 1440 ∈ wcel 2128 Ⅎwnfc 2286 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 (class class class)co 5825 Fincfn 6686 ℂcc 7731 · cmul 7738 ∏cprod 11451 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-ihash 10654 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-proddc 11452 |
This theorem is referenced by: fprodap0f 11537 fprodle 11541 |
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