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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 12985 | . . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) | |
2 | fprodm1.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
3 | eluzelz 12247 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
5 | 4 | zcnd 12082 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | 1cnd 10630 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 10995 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | eleq1d 2897 | . . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1)))) |
9 | 1, 8 | mtbii 328 | . . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) |
10 | disjsn 4641 | . . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1))) | |
11 | 9, 10 | sylibr 236 | . . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
12 | eluzel2 12242 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | peano2zm 12019 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
16 | 13 | zcnd 12082 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
17 | 16, 6 | npcand 10995 | . . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
18 | 17 | fveq2d 6669 | . . . . . . 7 ⊢ (𝜑 → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
19 | 2, 18 | eleqtrrd 2916 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) |
20 | eluzp1m1 12262 | . . . . . 6 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) | |
21 | 15, 19, 20 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) |
22 | fzsuc2 12959 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
23 | 13, 21, 22 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
24 | 7 | oveq2d 7166 | . . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁)) |
25 | 7 | sneqd 4573 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
26 | 25 | uneq2d 4139 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
27 | 23, 24, 26 | 3eqtr3d 2864 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
28 | fzfid 13335 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
29 | fprodm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
30 | 11, 27, 28, 29 | fprodsplit 15314 | . 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 2897 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
33 | 29 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
34 | eluzfz2 12909 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
35 | 2, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
36 | 32, 33, 35 | rspcdva 3625 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 31 | prodsn 15310 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
38 | 2, 36, 37 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵) |
39 | 38 | oveq2d 7166 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵)) |
40 | 30, 39 | eqtrd 2856 | 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 398 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4561 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 ∏cprod 15253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-prod 15254 |
This theorem is referenced by: fprodp1 15317 fprodm1s 15318 risefacp1 15377 fallfacp1 15378 prmop1 16368 bcprod 32965 |
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