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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 4711 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
| 4 | 2, 3 | sylibr 234 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
| 5 | eqidd 2738 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
| 6 | fprodsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 7 | snfi 9083 | . . . 4 ⊢ {𝐵} ∈ Fin | |
| 8 | unfi 9211 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
| 9 | 6, 7, 8 | sylancl 586 | . . 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 4154 | . . . . . . . 8 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ {𝐵}) | |
| 13 | elsni 4643 | . . . . . . . 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 2841 | . . . 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 16024 | . 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 16000 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
| 26 | 23, 18, 25 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷) |
| 27 | 26 | oveq2d 7447 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
| 28 | 22, 27 | eqtrd 2777 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 (class class class)co 7431 Fincfn 8985 ℂcc 11153 · cmul 11160 ∏cprod 15939 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 |
| This theorem is referenced by: fprodmodd 16033 coprmprod 16698 coprmproddvdslem 16699 breprexplema 34645 breprexplemc 34647 circlemethhgt 34658 fprodexp 45609 fprodabs2 45610 mccllem 45612 fprodcnlem 45614 fprodcncf 45915 dvmptfprodlem 45959 dvnprodlem2 45962 hspmbllem1 46641 |
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