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Mirrors > Home > MPE Home > Th. List > fprodsplitsn | Structured version Visualization version GIF version |
Description: Separate out a term in a finite product. (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 4521 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 226 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2780 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fprodsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | snfi 8391 | . . . 4 ⊢ {𝐵} ∈ Fin | |
8 | unfi 8580 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 577 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
10 | fprodsplitsn.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
11 | 10 | adantlr 702 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
12 | elunnel1 4016 | . . . . . . . 8 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ {𝐵}) | |
13 | elsni 4458 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
15 | 14 | adantll 701 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
16 | fprodsplitsn.d | . . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) |
18 | fprodsplitsn.dcn | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
19 | 18 | ad2antrr 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐷 ∈ ℂ) |
20 | 17, 19 | eqeltrd 2867 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
21 | 11, 20 | pm2.61dan 800 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
22 | 1, 4, 5, 9, 21 | fprodsplitf 15202 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶)) |
23 | fprodsplitsn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
24 | fprodsplitsn.kd | . . . . 5 ⊢ Ⅎ𝑘𝐷 | |
25 | 24, 16 | prodsnf 15178 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
26 | 23, 18, 25 | syl2anc 576 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
27 | 26 | oveq2d 6992 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
28 | 22, 27 | eqtrd 2815 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2050 Ⅎwnfc 2917 ∪ cun 3828 ∩ cin 3829 ∅c0 4179 {csn 4441 (class class class)co 6976 Fincfn 8306 ℂcc 10333 · cmul 10340 ∏cprod 15119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-prod 15120 |
This theorem is referenced by: fprodmodd 15211 coprmprod 15861 coprmproddvdslem 15862 breprexplema 31546 breprexplemc 31548 circlemethhgt 31559 fprodexp 41304 fprodabs2 41305 mccllem 41307 fprodcnlem 41309 fprodcncf 41612 dvmptfprodlem 41657 dvnprodlem2 41660 hspmbllem1 42337 |
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