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Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodaddrecnncnvlem | Structured version Visualization version GIF version |
Description: The sequence 𝑆 of finite products, where every factor is added an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
fprodaddrecnncnvlem.k | ⊢ Ⅎ𝑘𝜑 |
fprodaddrecnncnvlem.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodaddrecnncnvlem.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodaddrecnncnvlem.s | ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
fprodaddrecnncnvlem.f | ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥)) |
fprodaddrecnncnvlem.g | ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) |
Ref | Expression |
---|---|
fprodaddrecnncnvlem | ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11599 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 11285 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | fprodaddrecnncnvlem.k | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
4 | fprodaddrecnncnvlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | fprodaddrecnncnvlem.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
6 | fprodaddrecnncnvlem.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥)) | |
7 | 3, 4, 5, 6 | fprodadd2cncf 38793 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
8 | 1rp 11712 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ+) |
10 | nnrp 11718 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
11 | 9, 10 | rpdivcld 11765 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+) |
12 | 11 | rpcnd 11750 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℂ) |
13 | 12 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℂ) |
14 | fprodaddrecnncnvlem.g | . . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) | |
15 | 13, 14 | fmptd 6292 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ) |
16 | 1cnd 9935 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | divcnv 14424 | . . . . 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 4593 | . . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)) |
21 | 18, 20 | mpbird 246 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) |
22 | 0cnd 9912 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
23 | 1, 2, 7, 15, 21, 22 | climcncf 22511 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘0)) |
24 | nfv 1830 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℂ | |
25 | 3, 24 | nfan 1816 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℂ) |
26 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ Fin) |
27 | 5 | adantlr 747 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simplr 788 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℂ) | |
29 | 27, 28 | addcld 9938 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) ∈ ℂ) |
30 | 25, 26, 29 | fprodclf 14562 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) ∈ ℂ) |
31 | 30, 6 | fmptd 6292 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
32 | fcompt 6306 | . . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) | |
33 | 31, 15, 32 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
34 | fprodaddrecnncnvlem.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 6200 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (1 / 𝑛) ∈ ℂ) → (𝐺‘𝑛) = (1 / 𝑛)) |
38 | 36, 12, 37 | syl2anc 691 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (1 / 𝑛)) |
39 | 38 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
40 | 39 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
41 | 6 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))) |
42 | oveq2 6557 | . . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑛) → (𝐵 + 𝑥) = (𝐵 + (1 / 𝑛))) | |
43 | 42 | prodeq2ad 38659 | . . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑛) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
44 | 43 | adantl 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 = (1 / 𝑛)) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
45 | prodex 14476 | . . . . . . . . 9 ⊢ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)) ∈ V | |
46 | 45 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)) ∈ V) |
47 | 41, 44, 13, 46 | fvmptd 6197 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(1 / 𝑛)) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
48 | 40, 47 | eqtr2d 2645 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)) = (𝐹‘(𝐺‘𝑛))) |
49 | 48 | mpteq2dva 4672 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
50 | 35, 49 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
51 | 33, 50 | eqtr4d 2647 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = 𝑆) |
52 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))) |
53 | nfv 1830 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = 0 | |
54 | 3, 53 | nfan 1816 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 = 0) |
55 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 + 𝑥) = (𝐵 + 0)) | |
56 | 55 | ad2antlr 759 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = (𝐵 + 0)) |
57 | 5 | addid1d 10115 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
58 | 57 | adantlr 747 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
59 | 56, 58 | eqtrd 2644 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = 𝐵) |
60 | 59 | ex 449 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑘 ∈ 𝐴 → (𝐵 + 𝑥) = 𝐵)) |
61 | 54, 60 | ralrimi 2940 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 + 𝑥) = 𝐵) |
62 | 61 | prodeq2d 14491 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 𝐵) |
63 | prodex 14476 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ V) |
65 | 52, 62, 22, 64 | fvmptd 6197 | . . 3 ⊢ (𝜑 → (𝐹‘0) = ∏𝑘 ∈ 𝐴 𝐵) |
66 | 51, 65 | breq12d 4596 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ⇝ (𝐹‘0) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)) |
67 | 23, 66 | mpbid 221 | 1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 / cdiv 10563 ℕcn 10897 ℝ+crp 11708 ⇝ cli 14063 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-prod 14475 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 |
This theorem is referenced by: fprodaddrecnncnv 38797 |
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