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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 12293 | . . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) | |
2 | fprodm1.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
3 | eluzelz 11573 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
5 | 4 | zcnd 11359 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | 1cnd 9935 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 10275 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | eleq1d 2672 | . . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1)))) |
9 | 1, 8 | mtbii 315 | . . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) |
10 | disjsn 4192 | . . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) | |
11 | 9, 10 | sylibr 223 | . . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
12 | eluzel2 11568 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | peano2zm 11297 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
16 | 13 | zcnd 11359 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
17 | 16, 6 | npcand 10275 | . . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
18 | 17 | fveq2d 6107 | . . . . . . 7 ⊢ (𝜑 → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
19 | 2, 18 | eleqtrrd 2691 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) |
20 | eluzp1m1 11587 | . . . . . 6 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) | |
21 | 15, 19, 20 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) |
22 | fzsuc2 12268 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
23 | 13, 21, 22 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
24 | 7 | oveq2d 6565 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁)) |
25 | 7 | sneqd 4137 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
26 | 25 | uneq2d 3729 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
27 | 23, 24, 26 | 3eqtr3d 2652 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
28 | fzfid 12634 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
29 | fprodm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
30 | 11, 27, 28, 29 | fprodsplit 14535 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴)) |
31 | eluzfz2 12220 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
32 | 2, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
33 | 29 | ralrimiva 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
34 | fprodm1.3 | . . . . . . 7 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
35 | 34 | eleq1d 2672 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
36 | 35 | rspcv 3278 | . . . . 5 ⊢ (𝑁 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → 𝐵 ∈ ℂ)) |
37 | 32, 33, 36 | sylc 63 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
38 | 34 | prodsn 14531 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
39 | 2, 37, 38 | syl2anc 691 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
40 | 39 | oveq2d 6565 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵)) |
41 | 30, 40 | eqtrd 2644 | 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 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 |
This theorem is referenced by: fprodp1 14538 fprodm1s 14539 risefacp1 14599 fallfacp1 14600 prmop1 15580 bcprod 30877 |
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