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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 9934 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
3 | 1, 2 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
4 | elfzelz 9929 | . . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
5 | 3, 4 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | fzsn 9969 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
7 | 5, 6 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
8 | 7 | ineq1d 3307 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁))) |
9 | 5 | zred 9287 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
10 | 9 | ltp1d 8802 | . . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
11 | fzdisj 9955 | . . . . 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 2192 | . . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
14 | fzsplit 9954 | . . . . 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 3260 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
17 | 15, 16 | eqtrd 2190 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
18 | eluzelz 9449 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
19 | 1, 18 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | 5, 19 | fzfigd 10334 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
21 | elfzelz 9929 | . . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ) | |
22 | zdceq 9240 | . . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑗 = 𝑀) | |
23 | 21, 5, 22 | syl2anr 288 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 = 𝑀) |
24 | velsn 3577 | . . . . . 6 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀) | |
25 | 24 | dcbii 826 | . . . . 5 ⊢ (DECID 𝑗 ∈ {𝑀} ↔ DECID 𝑗 = 𝑀) |
26 | 23, 25 | sylibr 133 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ {𝑀}) |
27 | 26 | ralrimiva 2530 | . . 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 11497 | . 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 2226 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
32 | 28 | ralrimiva 2530 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
33 | 31, 32, 3 | rspcdva 2821 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
34 | 30 | prodsn 11494 | . . . 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 5840 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
37 | 29, 36 | eqtrd 2190 | 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 820 = wceq 1335 ∈ wcel 2128 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 class class class wbr 3966 ‘cfv 5171 (class class class)co 5825 ℂcc 7731 1c1 7734 + caddc 7736 · cmul 7738 < clt 7913 ℤcz 9168 ℤ≥cuz 9440 ...cfz 9913 ∏cprod 11451 |
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 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
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 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-ihash 10654 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-proddc 11452 |
This theorem is referenced by: (None) |
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