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Mirrors > Home > MPE Home > Th. List > fprodm1 | Structured version Visualization version GIF version |
Description: Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.) |
Ref | Expression |
---|---|
fprodm1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fprodm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fprodm1.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fprodm1 | ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzp1nel 13084 | . . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) | |
2 | fprodm1.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
3 | eluzelz 12336 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
5 | 4 | zcnd 12171 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | 1cnd 10716 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 11081 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | eleq1d 2817 | . . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1)))) |
9 | 1, 8 | mtbii 329 | . . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) |
10 | disjsn 4602 | . . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) | |
11 | 9, 10 | sylibr 237 | . . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
12 | eluzel2 12331 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | peano2zm 12108 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
16 | 13 | zcnd 12171 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
17 | 16, 6 | npcand 11081 | . . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
18 | 17 | fveq2d 6680 | . . . . . . 7 ⊢ (𝜑 → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
19 | 2, 18 | eleqtrrd 2836 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) |
20 | eluzp1m1 12352 | . . . . . 6 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) | |
21 | 15, 19, 20 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) |
22 | fzsuc2 13058 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
23 | 13, 21, 22 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
24 | 7 | oveq2d 7188 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁)) |
25 | 7 | sneqd 4528 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
26 | 25 | uneq2d 4053 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
27 | 23, 24, 26 | 3eqtr3d 2781 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
28 | fzfid 13434 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
29 | fprodm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
30 | 11, 27, 28, 29 | fprodsplit 15414 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴)) |
31 | fprodm1.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
32 | 31 | eleq1d 2817 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
33 | 29 | ralrimiva 3096 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
34 | eluzfz2 13008 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
35 | 2, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
36 | 32, 33, 35 | rspcdva 3528 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 31 | prodsn 15410 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
38 | 2, 36, 37 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
39 | 38 | oveq2d 7188 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵)) |
40 | 30, 39 | eqtrd 2773 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 ∩ cin 3842 ∅c0 4211 {csn 4516 ‘cfv 6339 (class class class)co 7172 ℂcc 10615 1c1 10618 + caddc 10620 · cmul 10622 − cmin 10950 ℤcz 12064 ℤ≥cuz 12326 ...cfz 12983 ∏cprod 15353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-sup 8981 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-n0 11979 df-z 12065 df-uz 12327 df-rp 12475 df-fz 12984 df-fzo 13127 df-seq 13463 df-exp 13524 df-hash 13785 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-prod 15354 |
This theorem is referenced by: fprodp1 15417 fprodm1s 15418 risefacp1 15477 fallfacp1 15478 prmop1 16476 bcprod 33279 aks4d1p1 39725 |
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