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Mathbox for Glauco Siliprandi |
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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 12093 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 11824 | . . 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 41652 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
8 | 1rp 12206 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ+) |
10 | nnrp 12215 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
11 | 9, 10 | rpdivcld 12263 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+) |
12 | 11 | rpcnd 12248 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℂ) |
13 | 12 | adantl 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℂ) |
14 | fprodaddrecnncnvlem.g | . . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) | |
15 | 13, 14 | fmptd 6699 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ) |
16 | 1cnd 10432 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | divcnv 15066 | . . . . 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 4935 | . . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)) |
21 | 18, 20 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) |
22 | 0cnd 10430 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
23 | 1, 2, 7, 15, 21, 22 | climcncf 23226 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘0)) |
24 | nfv 1874 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℂ | |
25 | 3, 24 | nfan 1863 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℂ) |
26 | 4 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ Fin) |
27 | 5 | adantlr 703 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simplr 757 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℂ) | |
29 | 27, 28 | addcld 10457 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) ∈ ℂ) |
30 | 25, 26, 29 | fprodclf 15204 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) ∈ ℂ) |
31 | 30, 6 | fmptd 6699 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
32 | fcompt 6716 | . . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) | |
33 | 31, 15, 32 | syl2anc 576 | . . . 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 6603 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (1 / 𝑛) ∈ ℂ) → (𝐺‘𝑛) = (1 / 𝑛)) |
38 | 36, 12, 37 | syl2anc 576 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (1 / 𝑛)) |
39 | 38 | fveq2d 6500 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
40 | 39 | adantl 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
41 | oveq2 6982 | . . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑛) → (𝐵 + 𝑥) = (𝐵 + (1 / 𝑛))) | |
42 | 41 | prodeq2ad 41336 | . . . . . . . 8 ⊢ (𝑥 = (1 / 𝑛) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
43 | prodex 15119 | . . . . . . . . 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 6615 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(1 / 𝑛)) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
46 | 40, 45 | eqtr2d 2808 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)) = (𝐹‘(𝐺‘𝑛))) |
47 | 46 | mpteq2dva 5018 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
48 | 35, 47 | eqtrd 2807 | . . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
49 | 33, 48 | eqtr4d 2810 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = 𝑆) |
50 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))) |
51 | nfv 1874 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = 0 | |
52 | 3, 51 | nfan 1863 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 = 0) |
53 | oveq2 6982 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 + 𝑥) = (𝐵 + 0)) | |
54 | 53 | ad2antlr 715 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = (𝐵 + 0)) |
55 | 5 | addid1d 10638 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
56 | 55 | adantlr 703 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
57 | 54, 56 | eqtrd 2807 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = 𝐵) |
58 | 57 | ex 405 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑘 ∈ 𝐴 → (𝐵 + 𝑥) = 𝐵)) |
59 | 52, 58 | ralrimi 3159 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 + 𝑥) = 𝐵) |
60 | 59 | prodeq2d 15134 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 𝐵) |
61 | prodex 15119 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ V) |
63 | 50, 60, 22, 62 | fvmptd 6599 | . . 3 ⊢ (𝜑 → (𝐹‘0) = ∏𝑘 ∈ 𝐴 𝐵) |
64 | 49, 63 | breq12d 4938 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ⇝ (𝐹‘0) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)) |
65 | 23, 64 | mpbid 224 | 1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 Ⅎwnf 1747 ∈ wcel 2051 Vcvv 3408 class class class wbr 4925 ↦ cmpt 5004 ∘ ccom 5407 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Fincfn 8304 ℂcc 10331 0cc0 10333 1c1 10334 + caddc 10336 / cdiv 11096 ℕcn 11437 ℝ+crp 12202 ⇝ cli 14700 ∏cprod 15117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-rlim 14705 df-prod 15118 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cn 21554 df-cnp 21555 df-tx 21889 df-hmeo 22082 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 |
This theorem is referenced by: fprodaddrecnncnv 41656 |
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