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Mirrors > Home > ILE Home > Th. List > fprod1p | GIF version |
Description: Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.) |
Ref | Expression |
---|---|
fprod1p.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fprod1p.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fprod1p.3 | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fprod1p | ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprod1p.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzfz1 9987 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
3 | 1, 2 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
4 | elfzelz 9981 | . . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
5 | 3, 4 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | fzsn 10022 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
7 | 5, 6 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
8 | 7 | ineq1d 3327 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁))) |
9 | 5 | zred 9334 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
10 | 9 | ltp1d 8846 | . . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
11 | fzdisj 10008 | . . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) | |
12 | 10, 11 | syl 14 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
13 | 8, 12 | eqtr3d 2205 | . . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
14 | fzsplit 10007 | . . . . 5 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
15 | 3, 14 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
16 | 7 | uneq1d 3280 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
17 | 15, 16 | eqtrd 2203 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
18 | eluzelz 9496 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
19 | 1, 18 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | 5, 19 | fzfigd 10387 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
21 | elfzelz 9981 | . . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ) | |
22 | zdceq 9287 | . . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑗 = 𝑀) | |
23 | 21, 5, 22 | syl2anr 288 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 = 𝑀) |
24 | velsn 3600 | . . . . . 6 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀) | |
25 | 24 | dcbii 835 | . . . . 5 ⊢ (DECID 𝑗 ∈ {𝑀} ↔ DECID 𝑗 = 𝑀) |
26 | 23, 25 | sylibr 133 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ {𝑀}) |
27 | 26 | ralrimiva 2543 | . . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)DECID 𝑗 ∈ {𝑀}) |
28 | fprod1p.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
29 | 13, 17, 20, 27, 28 | fprodsplitdc 11559 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
30 | fprod1p.3 | . . . . . 6 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
31 | 30 | eleq1d 2239 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
32 | 28 | ralrimiva 2543 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
33 | 31, 32, 3 | rspcdva 2839 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
34 | 30 | prodsn 11556 | . . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵) |
35 | 3, 33, 34 | syl2anc 409 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵) |
36 | 35 | oveq1d 5868 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
37 | 29, 36 | eqtrd 2203 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ∪ cun 3119 ∩ cin 3120 ∅c0 3414 {csn 3583 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 1c1 7775 + caddc 7777 · cmul 7779 < clt 7954 ℤcz 9212 ℤ≥cuz 9487 ...cfz 9965 ∏cprod 11513 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 |
This theorem is referenced by: (None) |
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