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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 12919 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12646 | . . 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 45862 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
8 | 1rp 13036 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ+) |
10 | nnrp 13044 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
11 | 9, 10 | rpdivcld 13092 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+) |
12 | 11 | rpcnd 13077 | . . . . 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 7134 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ) |
16 | 1cnd 11254 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | divcnv 15886 | . . . . 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 5158 | . . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)) |
21 | 18, 20 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) |
22 | 0cnd 11252 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
23 | 1, 2, 7, 15, 21, 22 | climcncf 24940 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘0)) |
24 | nfv 1912 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℂ | |
25 | 3, 24 | nfan 1897 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℂ) |
26 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ Fin) |
27 | 5 | adantlr 715 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simplr 769 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℂ) | |
29 | 27, 28 | addcld 11278 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) ∈ ℂ) |
30 | 25, 26, 29 | fprodclf 16025 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) ∈ ℂ) |
31 | 30, 6 | fmptd 7134 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
32 | fcompt 7153 | . . . . 5 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → (𝐹 ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) | |
33 | 31, 15, 32 | syl2anc 584 | . . . 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 7027 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (1 / 𝑛) ∈ ℂ) → (𝐺‘𝑛) = (1 / 𝑛)) |
38 | 36, 12, 37 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (1 / 𝑛)) |
39 | 38 | fveq2d 6911 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
40 | 39 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝐺‘𝑛)) = (𝐹‘(1 / 𝑛))) |
41 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑛) → (𝐵 + 𝑥) = (𝐵 + (1 / 𝑛))) | |
42 | 41 | prodeq2ad 45548 | . . . . . . . 8 ⊢ (𝑥 = (1 / 𝑛) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
43 | prodex 15938 | . . . . . . . . 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 7039 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘(1 / 𝑛)) = ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) |
46 | 40, 45 | eqtr2d 2776 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)) = (𝐹‘(𝐺‘𝑛))) |
47 | 46 | mpteq2dva 5248 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
48 | 35, 47 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ ℕ ↦ (𝐹‘(𝐺‘𝑛)))) |
49 | 33, 48 | eqtr4d 2778 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = 𝑆) |
50 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))) |
51 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = 0 | |
52 | 3, 51 | nfan 1897 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 = 0) |
53 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 + 𝑥) = (𝐵 + 0)) | |
54 | 53 | ad2antlr 727 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = (𝐵 + 0)) |
55 | 5 | addridd 11459 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
56 | 55 | adantlr 715 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 0) = 𝐵) |
57 | 54, 56 | eqtrd 2775 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝑥) = 𝐵) |
58 | 57 | ex 412 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑘 ∈ 𝐴 → (𝐵 + 𝑥) = 𝐵)) |
59 | 52, 58 | ralrimi 3255 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 + 𝑥) = 𝐵) |
60 | 59 | prodeq2d 15954 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0) → ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥) = ∏𝑘 ∈ 𝐴 𝐵) |
61 | prodex 15938 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ V) |
63 | 50, 60, 22, 62 | fvmptd 7023 | . . 3 ⊢ (𝜑 → (𝐹‘0) = ∏𝑘 ∈ 𝐴 𝐵) |
64 | 49, 63 | breq12d 5161 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ⇝ (𝐹‘0) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)) |
65 | 23, 64 | mpbid 232 | 1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 / cdiv 11918 ℕcn 12264 ℝ+crp 13032 ⇝ cli 15517 ∏cprod 15936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-prod 15937 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 |
This theorem is referenced by: fprodaddrecnncnv 45866 |
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