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Mirrors > Home > MPE Home > Th. List > zprodn0 | Structured version Visualization version 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) |
zprodn0.3 | ⊢ (𝜑 → 𝑋 ≠ 0) |
zprodn0.4 | ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
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 |
---|---|
zprodn0 | ⊢ (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 | zprodn0.3 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
5 | 1, 2, 3, 4 | ntrivcvgn0 15239 | . . 3 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑥(𝑥 ≠ 0 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) |
6 | zprodn0.5 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
7 | zprodn0.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) | |
8 | zprodn0.7 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
9 | 1, 2, 5, 6, 7, 8 | zprod 15276 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹))) |
10 | fclim 14895 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
11 | ffun 6503 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
13 | funbrfv 6702 | . . 3 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)) | |
14 | 12, 3, 13 | mpsyl 68 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
15 | 9, 14 | eqtrd 2856 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3924 ifcif 4453 class class class wbr 5052 dom cdm 5541 Fun wfun 6335 ⟶wf 6337 ‘cfv 6341 ℂcc 10521 0cc0 10523 1c1 10524 · cmul 10528 ℤcz 11968 ℤ≥cuz 12230 seqcseq 13359 ⇝ cli 14826 ∏cprod 15244 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-prod 15245 |
This theorem is referenced by: iprodn0 15279 prod0 15282 prod1 15283 |
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