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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodsubrecnncnvlem | Structured version Visualization version GIF version |
Description: The sequence 𝑆 of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
fprodsubrecnncnvlem.k | ⊢ Ⅎ𝑘𝜑 |
fprodsubrecnncnvlem.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodsubrecnncnvlem.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodsubrecnncnvlem.s | ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) |
fprodsubrecnncnvlem.f | ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥)) |
fprodsubrecnncnvlem.g | ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) |
Ref | Expression |
---|---|
fprodsubrecnncnvlem | ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12269 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12001 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | fprodsubrecnncnvlem.k | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
4 | fprodsubrecnncnvlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fprodsubrecnncnvlem.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
6 | fprodsubrecnncnvlem.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥)) | |
7 | 3, 4, 5, 6 | fprodsub2cncf 42547 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
8 | 1rp 12381 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ+) |
10 | nnrp 12388 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
11 | 9, 10 | rpdivcld 12436 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+) |
12 | 11 | rpcnd 12421 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℂ) |
13 | 12 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℂ) |
14 | fprodsubrecnncnvlem.g | . . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) | |
15 | 13, 14 | fmptd 6855 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ) |
16 | 1cnd 10625 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | divcnv 15200 | . . . . 5 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0) |
19 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))) |
20 | 19 | breq1d 5040 | . . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)) |
21 | 18, 20 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) |
22 | 0cnd 10623 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
23 | 1, 2, 7, 15, 21, 22 | climcncf 23505 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘0)) |
24 | nfv 1915 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℂ | |
25 | 3, 24 | nfan 1900 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℂ) |
26 | 4 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ Fin) |
27 | 5 | adantlr 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simplr 768 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℂ) | |
29 | 27, 28 | subcld 10986 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) ∈ ℂ) |
30 | 25, 26, 29 | fprodclf 15338 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) ∈ ℂ) |
31 | 30, 6 | fmptd 6855 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
32 | fcompt 6872 | . . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) | |
33 | 31, 15, 32 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
34 | fprodsubrecnncnvlem.s | . . . . . 6 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)))) |
36 | id 22 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ) | |
37 | 14 | fvmpt2 6756 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (1 / 𝑛) ∈ ℂ) → (𝐺‘𝑛) = (1 / 𝑛)) |
38 | 36, 12, 37 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (1 / 𝑛)) |
39 | 38 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
40 | 39 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
41 | oveq2 7143 | . . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑛) → (𝐵 − 𝑥) = (𝐵 − (1 / 𝑛))) | |
42 | 41 | prodeq2ad 42234 | . . . . . . . 8 ⊢ (𝑥 = (1 / 𝑛) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) |
43 | prodex 15253 | . . . . . . . . 9 ⊢ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)) ∈ V | |
44 | 43 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)) ∈ V) |
45 | 6, 42, 13, 44 | fvmptd3 6768 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(1 / 𝑛)) = ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) |
46 | 40, 45 | eqtr2d 2834 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)) = (𝐹‘(𝐺‘𝑛))) |
47 | 46 | mpteq2dva 5125 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
48 | 35, 47 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
49 | 33, 48 | eqtr4d 2836 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = 𝑆) |
50 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥))) |
51 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = 0 | |
52 | 3, 51 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 = 0) |
53 | oveq2 7143 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 − 𝑥) = (𝐵 − 0)) | |
54 | 53 | ad2antlr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) = (𝐵 − 0)) |
55 | 5 | subid1d 10975 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 − 0) = 𝐵) |
56 | 55 | adantlr 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 0) = 𝐵) |
57 | 54, 56 | eqtrd 2833 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) = 𝐵) |
58 | 57 | ex 416 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑘 ∈ 𝐴 → (𝐵 − 𝑥) = 𝐵)) |
59 | 52, 58 | ralrimi 3180 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 − 𝑥) = 𝐵) |
60 | 59 | prodeq2d 15268 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) = ∏𝑘 ∈ 𝐴 𝐵) |
61 | prodex 15253 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ V) |
63 | 50, 60, 22, 62 | fvmptd 6752 | . . 3 ⊢ (𝜑 → (𝐹‘0) = ∏𝑘 ∈ 𝐴 𝐵) |
64 | 49, 63 | breq12d 5043 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ⇝ (𝐹‘0) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)) |
65 | 23, 64 | mpbid 235 | 1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 − cmin 10859 / cdiv 11286 ℕcn 11625 ℝ+crp 12377 ⇝ cli 14833 ∏cprod 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-prod 15252 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 |
This theorem is referenced by: fprodsubrecnncnv 42550 |
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