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Mirrors > Home > MPE Home > Th. List > Mathboxes > prodsplit | Structured version Visualization version GIF version |
Description: Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.) |
Ref | Expression |
---|---|
prodsplit.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
prodsplit.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
prodsplit.3 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
prodsplit.4 | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
prodsplit.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
prodsplit | ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodsplit.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
2 | 1 | zred 12081 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | 2 | ltp1d 11563 | . . 3 ⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
4 | fzdisj 12931 | . . 3 ⊢ (𝑁 < (𝑁 + 1) → ((𝑀...𝑁) ∩ ((𝑁 + 1)...(𝑁 + 𝐾))) = ∅) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝑀...𝑁) ∩ ((𝑁 + 1)...(𝑁 + 𝐾))) = ∅) |
6 | prodsplit.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | prodsplit.4 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
8 | 7 | nn0zd 12079 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
9 | 1, 8 | zaddcld 12085 | . . . 4 ⊢ (𝜑 → (𝑁 + 𝐾) ∈ ℤ) |
10 | prodsplit.3 | . . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | |
11 | nn0addge1 11937 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝐾 ∈ ℕ0) → 𝑁 ≤ (𝑁 + 𝐾)) | |
12 | 2, 7, 11 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 𝐾)) |
13 | elfz4 12898 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ (𝑁 + 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ (𝑁 + 𝐾))) → 𝑁 ∈ (𝑀...(𝑁 + 𝐾))) | |
14 | 6, 9, 1, 10, 12, 13 | syl32anc 1373 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑀...(𝑁 + 𝐾))) |
15 | fzsplit 12930 | . . 3 ⊢ (𝑁 ∈ (𝑀...(𝑁 + 𝐾)) → (𝑀...(𝑁 + 𝐾)) = ((𝑀...𝑁) ∪ ((𝑁 + 1)...(𝑁 + 𝐾)))) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝑀...(𝑁 + 𝐾)) = ((𝑀...𝑁) ∪ ((𝑁 + 1)...(𝑁 + 𝐾)))) |
17 | fzfid 13338 | . 2 ⊢ (𝜑 → (𝑀...(𝑁 + 𝐾)) ∈ Fin) | |
18 | prodsplit.5 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ) | |
19 | 5, 16, 17, 18 | fprodsplit 15313 | 1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∪ cun 3927 ∩ cin 3928 ∅c0 4284 class class class wbr 5059 (class class class)co 7149 ℂcc 10528 ℝcr 10529 1c1 10531 + caddc 10533 · cmul 10535 < clt 10668 ≤ cle 10669 ℕ0cn0 11891 ℤcz 11975 ...cfz 12889 ∏cprod 15252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-clim 14838 df-prod 15253 |
This theorem is referenced by: (None) |
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