Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fprod1p | Structured version Visualization version 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 12917 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
4 | elfzelz 12911 | . . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | fzsn 12952 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
8 | 7 | ineq1d 4190 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁))) |
9 | 5 | zred 12090 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
10 | 9 | ltp1d 11572 | . . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
11 | fzdisj 12937 | . . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
13 | 8, 12 | eqtr3d 2860 | . . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) |
14 | fzsplit 12936 | . . . . 5 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
15 | 3, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
16 | 7 | uneq1d 4140 | . . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
17 | 15, 16 | eqtrd 2858 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
18 | fzfid 13344 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
19 | fprod1p.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
20 | 13, 17, 18, 19 | fprodsplit 15322 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
21 | fprod1p.3 | . . . . . 6 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
22 | 21 | eleq1d 2899 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
23 | 19 | ralrimiva 3184 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
24 | 22, 23, 3 | rspcdva 3627 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
25 | 21 | prodsn 15318 | . . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵) |
26 | 3, 24, 25 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵) |
27 | 26 | oveq1d 7173 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
28 | 20, 27 | eqtrd 2858 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ℤcz 11984 ℤ≥cuz 12246 ...cfz 12895 ∏cprod 15261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 |
This theorem is referenced by: fallfacfwd 15392 0fallfac 15393 etransclem4 42530 etransclem31 42557 etransclem35 42561 |
Copyright terms: Public domain | W3C validator |