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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 12282 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12014 | . . 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 42209 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
8 | 1rp 12394 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ+) |
10 | nnrp 12401 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
11 | 9, 10 | rpdivcld 12449 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+) |
12 | 11 | rpcnd 12434 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℂ) |
13 | 12 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℂ) |
14 | fprodsubrecnncnvlem.g | . . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) | |
15 | 13, 14 | fmptd 6878 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ) |
16 | 1cnd 10636 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | divcnv 15208 | . . . . 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 5076 | . . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)) |
21 | 18, 20 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) |
22 | 0cnd 10634 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
23 | 1, 2, 7, 15, 21, 22 | climcncf 23508 | . 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 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ Fin) |
27 | 5 | adantlr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simplr 767 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℂ) | |
29 | 27, 28 | subcld 10997 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) ∈ ℂ) |
30 | 25, 26, 29 | fprodclf 15346 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) ∈ ℂ) |
31 | 30, 6 | fmptd 6878 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
32 | fcompt 6895 | . . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) | |
33 | 31, 15, 32 | syl2anc 586 | . . . 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 6779 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (1 / 𝑛) ∈ ℂ) → (𝐺‘𝑛) = (1 / 𝑛)) |
38 | 36, 12, 37 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (1 / 𝑛)) |
39 | 38 | fveq2d 6674 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
40 | 39 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
41 | oveq2 7164 | . . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑛) → (𝐵 − 𝑥) = (𝐵 − (1 / 𝑛))) | |
42 | 41 | prodeq2ad 41893 | . . . . . . . 8 ⊢ (𝑥 = (1 / 𝑛) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) |
43 | prodex 15261 | . . . . . . . . 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 6791 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(1 / 𝑛)) = ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) |
46 | 40, 45 | eqtr2d 2857 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)) = (𝐹‘(𝐺‘𝑛))) |
47 | 46 | mpteq2dva 5161 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
48 | 35, 47 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
49 | 33, 48 | eqtr4d 2859 | . . 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 7164 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 − 𝑥) = (𝐵 − 0)) | |
54 | 53 | ad2antlr 725 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) = (𝐵 − 0)) |
55 | 5 | subid1d 10986 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 − 0) = 𝐵) |
56 | 55 | adantlr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 0) = 𝐵) |
57 | 54, 56 | eqtrd 2856 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝑥) = 𝐵) |
58 | 57 | ex 415 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑘 ∈ 𝐴 → (𝐵 − 𝑥) = 𝐵)) |
59 | 52, 58 | ralrimi 3216 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 − 𝑥) = 𝐵) |
60 | 59 | prodeq2d 15276 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥) = ∏𝑘 ∈ 𝐴 𝐵) |
61 | prodex 15261 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ V) |
63 | 50, 60, 22, 62 | fvmptd 6775 | . . 3 ⊢ (𝜑 → (𝐹‘0) = ∏𝑘 ∈ 𝐴 𝐵) |
64 | 49, 63 | breq12d 5079 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ⇝ (𝐹‘0) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)) |
65 | 23, 64 | mpbid 234 | 1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 0cc0 10537 1c1 10538 − cmin 10870 / cdiv 11297 ℕcn 11638 ℝ+crp 12390 ⇝ cli 14841 ∏cprod 15259 |
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 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-mulf 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 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-prod 15260 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 |
This theorem is referenced by: fprodsubrecnncnv 42212 |
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