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Mirrors > Home > ILE Home > Th. List > zprodap0 | GIF version |
Description: Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.) |
Ref | Expression |
---|---|
zprodn0.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
zprodn0.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
zprodap0.3 | ⊢ (𝜑 → 𝑋 # 0) |
zprodn0.4 | ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
zprodap0.dc | ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
zprodn0.5 | ⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
zprodn0.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
zprodn0.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
zprodap0 | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zprodn0.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | zprodn0.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | zprodn0.4 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
4 | zprodap0.3 | . . . 4 ⊢ (𝜑 → 𝑋 # 0) | |
5 | 1, 2, 3, 4 | ntrivcvgap0 11423 | . . 3 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑥(𝑥 # 0 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) |
6 | zprodn0.5 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
7 | zprodap0.dc | . . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) | |
8 | zprodn0.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) | |
9 | zprodn0.7 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
10 | 1, 2, 5, 6, 7, 8, 9 | zproddc 11453 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹))) |
11 | fclim 11168 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
12 | ffun 5315 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
14 | funbrfv 5500 | . . 3 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)) | |
15 | 13, 3, 14 | mpsyl 65 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
16 | 10, 15 | eqtrd 2187 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 820 = wceq 1332 ∈ wcel 2125 ∀wral 2432 ⊆ wss 3098 ifcif 3501 class class class wbr 3961 dom cdm 4579 Fun wfun 5157 ⟶wf 5159 ‘cfv 5163 ℂcc 7709 0cc0 7711 1c1 7712 · cmul 7716 # cap 8435 ℤcz 9146 ℤ≥cuz 9418 seqcseq 10322 ⇝ cli 11152 ∏cprod 11424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-oadd 6357 df-er 6469 df-en 6675 df-dom 6676 df-fin 6677 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-fz 9891 df-fzo 10020 df-seqfrec 10323 df-exp 10397 df-ihash 10627 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-proddc 11425 |
This theorem is referenced by: iprodap0 11456 prod0 11459 prod1dc 11460 |
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