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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 4617 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 237 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2735 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fprodsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | snfi 8710 | . . . 4 ⊢ {𝐵} ∈ Fin | |
8 | unfi 8839 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 589 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
10 | fprodsplitsn.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
11 | 10 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
12 | elunnel1 4054 | . . . . . . . 8 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ {𝐵}) | |
13 | elsni 4548 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
15 | 14 | adantll 714 | . . . . . 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 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐷 ∈ ℂ) |
20 | 17, 19 | eqeltrd 2834 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
21 | 11, 20 | pm2.61dan 813 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
22 | 1, 4, 5, 9, 21 | fprodsplitf 15531 | . 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 15507 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
26 | 23, 18, 25 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
27 | 26 | oveq2d 7218 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
28 | 22, 27 | eqtrd 2774 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2880 ∪ cun 3855 ∩ cin 3856 ∅c0 4227 {csn 4531 (class class class)co 7202 Fincfn 8615 ℂcc 10710 · cmul 10717 ∏cprod 15448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 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 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-prod 15449 |
This theorem is referenced by: fprodmodd 15540 coprmprod 16199 coprmproddvdslem 16200 breprexplema 32294 breprexplemc 32296 circlemethhgt 32307 fprodexp 42764 fprodabs2 42765 mccllem 42767 fprodcnlem 42769 fprodcncf 43070 dvmptfprodlem 43114 dvnprodlem2 43117 hspmbllem1 43793 |
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