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Mirrors > Home > ILE Home > Th. List > fprodsplit1f | GIF version |
Description: Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodsplit1f.kph | ⊢ Ⅎ𝑘𝜑 |
fprodsplit1f.fk | ⊢ (𝜑 → Ⅎ𝑘𝐷) |
fprodsplit1f.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodsplit1f.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodsplit1f.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
fprodsplit1f.d | ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
fprodsplit1f | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodsplit1f.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | disjdif 3466 | . . . 4 ⊢ ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅ | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝜑 → ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅) |
4 | fprodsplit1f.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fprodsplit1f.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
6 | snfig 6759 | . . . . 5 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ∈ Fin) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐶} ∈ Fin) |
8 | 5 | snssd 3701 | . . . 4 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
9 | undiffi 6869 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐶} ∈ Fin ∧ {𝐶} ⊆ 𝐴) → 𝐴 = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) | |
10 | 4, 7, 8, 9 | syl3anc 1220 | . . 3 ⊢ (𝜑 → 𝐴 = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) |
11 | fprodsplit1f.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
12 | 1, 3, 10, 4, 11 | fprodsplitf 11529 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
13 | 5 | ancli 321 | . . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
14 | nfv 1508 | . . . . . . . . 9 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
15 | 1, 14 | nfan 1545 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
16 | nfcsb1v 3064 | . . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 | |
17 | 16 | nfel1 2310 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
18 | 15, 17 | nfim 1552 | . . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
19 | eleq1 2220 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
20 | 19 | anbi2d 460 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
21 | csbeq1a 3040 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
22 | 21 | eleq1d 2226 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
23 | 20, 22 | imbi12d 233 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
24 | 18, 23, 11 | vtoclg1f 2771 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
25 | 5, 13, 24 | sylc 62 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
26 | prodsns 11500 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
27 | 5, 25, 26 | syl2anc 409 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) |
28 | fprodsplit1f.fk | . . . . 5 ⊢ (𝜑 → Ⅎ𝑘𝐷) | |
29 | fprodsplit1f.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) | |
30 | 1, 28, 5, 29 | csbiedf 3071 | . . . 4 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
31 | 27, 30 | eqtrd 2190 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = 𝐷) |
32 | 31 | oveq1d 5839 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
33 | 12, 32 | eqtrd 2190 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 Ⅎwnf 1440 ∈ wcel 2128 Ⅎwnfc 2286 ⦋csb 3031 ∖ cdif 3099 ∪ cun 3100 ∩ cin 3101 ⊆ wss 3102 ∅c0 3394 {csn 3560 (class class class)co 5824 Fincfn 6685 ℂcc 7730 · cmul 7737 ∏cprod 11447 |
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 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
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 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-ihash 10650 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-proddc 11448 |
This theorem is referenced by: fprodeq0g 11535 |
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